Rational Expressions and Equations

A rational expression is a fraction in which the numerator and the denominator are polynomials. A rational equation is an equation that contains at least one rational expression. Rational expressions can be simplified, added, subtracted, multiplied, or divided, just like regular fractions. The main difference is that with rational expressions, we must always consider the values that make the denominator zero, as they would make the expression undefined.

General Form of Rational Expression:

$\frac{P(x)}{Q(x)}$

Where:

• $P(x)$ is the numerator polynomial.
• $Q(x)$ is the denominator polynomial, and $Q(x)\ne 0$.

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Example 1:

Simplify the rational expression:

$\frac{2x^2+6x}{4x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{2x(x+3)}{4x}$

Step 2: Cancel out the common factor $x$.

$=\frac{2(x+3)}{4}$

Step 3: Simplify the constants.

$=\frac{x+3}{2}$

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Example 2:

Simplify the rational expression:

$\frac{x^2–9}{x^2–6x+9}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–3)(x+3)}{(x–3)(x–3)}$

Step 2: Cancel out the common factor $x–3$.

$=\frac{x+3}{x–3}$

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Example 3:

Simplify the rational expression:

$\frac{6x^2–12}{x^2+2x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{6(x^2–2)}{x(x+2)}$

Step 2: Simplify the common factors if any.

$=\frac{6(x–2)(x+2)}{x(x+2)}$

Step 3: Cancel out the common factor $x+2$.

$=\frac{6(x–2)}{x}$

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Example 4:

Add the rational expressions:

$\frac{1}{x}+\frac{1}{x+2}$

Solution:

Step 1: Find the LCD, which is $x(x+2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{(x+2)}{x(x+2)}+\frac{x}{x(x+2)}$

Step 3: Combine the numerators.

$=\frac{(x+2+x)}{x(x+2)}$

Step 4: Simplify the numerator.

$=\frac{2x+2}{x(x+2)}$

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Example 5:

Subtract the rational expressions:

$\frac{3}{x^2}–\frac{1}{x^3}$

Solution:

Step 1: Find the LCD, which is $x^3$.

Step 2: Rewrite each expression with the LCD.

$=\frac{3x}{x^3}–\frac{1}{x^3}$

Step 3: Combine the numerators.

$=\frac{(3x–1)}{x^3}$

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Example 6:

Add the rational expressions:

$\frac{2x}{x–3}+\frac{5x}{x+4}$

Solution:

Step 1: Find the LCD, which is $(x–3)(x+4)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{2x(x+4)}{(x–3)(x+4)}+\frac{5x(x–3)}{(x–3)(x+4)}$

Step 3: Combine the numerators.

$=\frac{2x(x+4)+5x(x–3)}{(x–3)(x+4)}$

Step 4: Simplify the numerator.

$=\frac{2x^2+8x+5x^2–15x}{(x–3)(x+4)}$

Step 5: Combine like terms.

$=\frac{7x^2–7x}{(x–3)(x+4)}$

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Example 7:

Multiply the rational expressions:

$\frac{x+3}{x–2}\times \frac{2x}{x+3}$

Solution:

Step 1: Multiply the numerators and the denominators.

$=\frac{(x+3)\times 2x}{(x–2)\times (x+3)}$

Step 2: Cancel out the common factor $x+3$.

$=\frac{2x}{x–2}$

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Example 8:

Divide the rational expressions:

$\frac{4x}{x^2–9}÷\frac{2x}{x–3}$

Solution:

Step 1: Rewrite the division as multiplication by the reciprocal.

$=\frac{4x}{(x–3)(x+3)}\times \frac{x–3}{2x}$

Step 2: Cancel the common factor $x–3$ and $x$.

$=\frac{4}{2(x+3)}$

Step 3: Simplify the constants.

$=\frac{2}{x+3}$

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Example 9:

Solve the rational equation:

$\frac{x+1}{x–2}=\frac{2}{x–2}$

Solution:

Step 1: Multiply both sides by $x–2$ to eliminate the denominator.

$(x+1)=2$

Step 2: Solve for $x$.

$x+1=2$

$x=1$

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Example 10:

Solve the rational equation:

$\frac{3}{x}–\frac{2}{x+1}=1$

Solution:

Step 1: Find the LCD, which is $x(x+1)$.

Step 2: Multiply both sides of the equation by the LCD.

$3(x+1)–2x=x(x+1)$

Step 3: Expand both sides.

$3x+3–2x=x^2+x$

Step 4: Combine like terms.

$x+3=x^2+x$

Step 5: Move all terms to one side.

$0=x^2–3$

Step 6: Solve for $x$.

$x^2=3$

$x=\pm \sqrt{3}$

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Example 11:

Solve the rational equation:

$\frac{5}{x+2}=\frac{3x}{x+2}$

Solution:

Step 1: Multiply both sides by $x+2$ to eliminate the denominator.

$5=3x$

Step 2: Solve for $x$.

$x=\frac{5}{3}$

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Example 12:

Solve the rational equation:

$\frac{2}{x}+\frac{3}{x+1}=\frac{5}{x}$

Solution:

Step 1: Find the LCD, which is $x(x+1)$.

Step 2: Multiply both sides of the equation by the LCD.

$2(x+1)+3x=5(x+1)$

Step 3: Expand both sides.

$2x+2+3x=5x+5$

Step 4: Combine like terms.

$5x+2=5x+5$

Step 5: Cancel $5x$ on both sides.

$2=5$

Since this is not possible, there is no solution.

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Example 13:

Solve the rational equation:

$\frac{2x}{x^2–1}=\frac{1}{x–1}$

Solution:

Step 1: Factor the denominator $x^2–1=(x–1)(x+1)$.

$=\frac{2x}{(x–1)(x+1)}=\frac{1}{x–1}$

Step 2: Multiply both sides by $(x–1)(x+1)$ to eliminate the denominators.

$2x=1(x+1)$

Step 3: Expand the right-hand side.

$2x=x+1$

Step 4: Move all terms to one side.

$2x–x=1$

$x=1$

Step 5: Check for extraneous solutions.

Since $x=1$ makes the denominator zero in the original equation, there is no solution.

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Example 14:

Simplify the rational expression:

$\frac{x^2+5x+6}{x^2+2x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x+2)(x+3)}{x(x+2)}$

Step 2: Cancel the common factor $(x+2)$.

$=\frac{x+3}{x}$

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Example 15:

Simplify the rational expression:

$\frac{2x^2–18}{x^2+2x–8}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{2(x^2–9)}{(x+4)(x–2)}$

Step 2: Factor $x^2–9$ as $(x+3)(x–3)$.

$=\frac{2(x+3)(x–3)}{(x+4)(x–2)}$

No common factors exist, so the simplified form is:

$=\frac{2(x+3)(x–3)}{(x+4)(x–2)}$

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Example 16:

Simplify the rational expression:

$\frac{3x^2+12x}{x^2–4}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{3x(x+4)}{(x–2)(x+2)}$

No common factors exist, so the simplified form is:

$=\frac{3x(x+4)}{(x–2)(x+2)}$

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Example 17:

Add the rational expressions:

$\frac{x+1}{x–2}+\frac{2x–3}{x+3}$

Solution:

Step 1: Find the LCD, which is $(x–2)(x+3)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{(x+1)(x+3)}{(x–2)(x+3)}+\frac{(2x–3)(x–2)}{(x–2)(x+3)}$

Step 3: Combine the numerators.

$=\frac{(x+1)(x+3)+(2x–3)(x–2)}{(x–2)(x+3)}$

Step 4: Expand the numerators.

$=\frac{x^2+3x+x+3+2x^2–4x–3x+6}{(x–2)(x+3)}$

Step 5: Simplify the numerator.

$=\frac{3x^2–3x+9}{(x–2)(x+3)}$

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Example 18:

Subtract the rational expressions:

$\frac{4x}{x–1}–\frac{3}{x+2}$

Solution:

Step 1: Find the LCD, which is $(x–1)(x+2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{4x(x+2)}{(x–1)(x+2)}–\frac{3(x–1)}{(x–1)(x+2)}$

Step 3: Combine the numerators.

$=\frac{4x(x+2)–3(x–1)}{(x–1)(x+2)}$

Step 4: Expand the numerators.

$=\frac{4x^2+8x–3x+3}{(x–1)(x+2)}$

Step 5: Simplify the numerator.

$=\frac{4x^2+5x+3}{(x–1)(x+2)}$

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Example 19:

Solve the rational equation:

$\frac{x+2}{x–3}=\frac{3x+5}{x+2}$

Solution:

Step 1: Cross multiply.

$(x+2)(x+2)=(x–3)(3x+5)$

Step 2: Expand both sides.

$x^2+4x+4=3x^2–9x+5x–15$

Step 3: Combine like terms.

$x^2+4x+4=3x^2–4x–15$

Step 4: Move all terms to one side.

$0=2x^2–8x–19$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-8)\pm \sqrt{(-8{)}^2–4(2)(-19)}}{2(2)}$

$x=\frac{8\pm \sqrt{64+152}}{4}$

$x=\frac{8\pm \sqrt{216}}{4}$

$x=\frac{8\pm 14.7}{4}$

Step 6: Solve for $x$.

$x=\frac{8+14.7}{4}=5.175$

$x=\frac{8–14.7}{4}=-1.675$

Thus, the solutions are $x=5.175$ and $x=-1.675$.

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Example 20:

Simplify the rational expression:

$\frac{6x^2–18}{x^2–9}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{6(x^2–3)}{(x–3)(x+3)}$

Step 2: Simplify.

$=\frac{6}{x+3}$

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Example 21:

Solve the rational equation:

$\frac{3x}{x+5}–\frac{2}{x–4}=\frac{7x}{x+5}$

Solution:

Step 1: Find the LCD, which is $(x+5)(x–4)$.

Step 2: Multiply both sides by the LCD.

$3x(x–4)–2(x+5)=7x(x–4)$

Step 3: Expand both sides.

$3x^2–12x–2x–10=7x^2–28x$

Step 4: Combine like terms.

$3x^2–14x–10=7x^2–28x$

Step 5: Move all terms to one side.

$3x^2–14x–10–7x^2+28x=0$

$-4x^2+14x–10=0$

Step 6: Solve the quadratic equation using the quadratic formula.

$x=\frac{-14\pm \sqrt{{14}^2–4(-4)(-10)}}{2(-4)}$

$x=\frac{-14\pm \sqrt{196–160}}{-8}$

$x=\frac{-14\pm \sqrt{36}}{-8}$

$x=\frac{-14\pm 6}{-8}$

Step 7: Solve for $x$.

$x=\frac{-14+6}{-8}=1$

$x=\frac{-14–6}{-8}=\frac{5}{2}$

Thus, the solutions are $x=1$ and $x=\frac{5}{2}$.

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Example 22:

Solve the rational equation:

$\frac{x+3}{x+2}=\frac{4}{x+5}$

Solution:

Step 1: Cross multiply.

$(x+3)(x+5)=4(x+2)$

Step 2: Expand both sides.

$x^2+5x+3x+15=4x+8$

Step 3: Combine like terms.

$x^2+8x+15=4x+8$

Step 4: Move all terms to one side.

$x^2+8x+15–4x–8=0$

$x^2+4x+7=0$

Step 5: Use the quadratic formula to solve for $x$.

$x=\frac{-4\pm \sqrt{4^2–4(1)(7)}}{2(1)}$

$x=\frac{-4\pm \sqrt{16–28}}{2}$

$x=\frac{-4\pm \sqrt{-12}}{2}$

Since this results in a negative discriminant, there is no real solution.

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Example 23:

Simplify the rational expression:

$\frac{3x^2+9x}{x^2+2x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{3x(x+3)}{x(x+2)}$

Step 2: Cancel the common factor $x$.

$=\frac{3(x+3)}{x+2}$

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Example 24:

Simplify the rational expression:

$\frac{x^2–4}{x^2+x–6}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–2)(x+2)}{(x+3)(x–2)}$

Step 2: Cancel the common factor $(x–2)$.

$=\frac{x+2}{x+3}$

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Example 25:

Add the rational expressions:

$\frac{5x}{x–1}+\frac{2}{x+3}$

Solution:

Step 1: Find the LCD, which is $(x–1)(x+3)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{5x(x+3)}{(x–1)(x+3)}+\frac{2(x–1)}{(x–1)(x+3)}$

Step 3: Combine the numerators.

$=\frac{5x(x+3)+2(x–1)}{(x–1)(x+3)}$

Step 4: Expand the numerators.

$=\frac{5x^2+15x+2x–2}{(x–1)(x+3)}$

Step 5: Combine like terms.

$=\frac{5x^2+17x–2}{(x–1)(x+3)}$

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Example 26:

Subtract the rational expressions:

$\frac{3}{x–4}–\frac{4}{x+5}$

Solution:

Step 1: Find the LCD, which is $(x–4)(x+5)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{3(x+5)}{(x–4)(x+5)}–\frac{4(x–4)}{(x–4)(x+5)}$

Step 3: Combine the numerators.

$=\frac{3(x+5)–4(x–4)}{(x–4)(x+5)}$

Step 4: Expand the numerators.

$=\frac{3x+15–4x+16}{(x–4)(x+5)}$

Step 5: Simplify.

$=\frac{-x+31}{(x–4)(x+5)}$

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Example 27:

Multiply the rational expressions:

$\frac{x+3}{x^2–1}\times \frac{2x}{x–1}$

Solution:

Step 1: Factor the denominator $x^2–1=(x–1)(x+1)$.

$=\frac{x+3}{(x–1)(x+1)}\times \frac{2x}{x–1}$

Step 2: Multiply the numerators and the denominators.

$=\frac{(x+3)\times 2x}{(x–1)(x+1)\times (x–1)}$

Step 3: Simplify.

$=\frac{2x(x+3)}{(x–1{)}^2(x+1)}$

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Example 28:

Divide the rational expressions:

$\frac{4x}{x^2–9}÷\frac{3x}{x–3}$

Solution:

Step 1: Rewrite the division as multiplication by the reciprocal.

$=\frac{4x}{(x–3)(x+3)}\times \frac{x–3}{3x}$

Step 2: Cancel the common factor $x–3$ and $x$.

$=\frac{4}{3(x+3)}$

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Example 29:

Solve the rational equation:

$\frac{x+2}{x–3}=\frac{3x+5}{x+2}$

Solution:

Step 1: Cross multiply.

$(x+2)(x+2)=(x–3)(3x+5)$

Step 2: Expand both sides.

$x^2+4x+4=3x^2–9x+5x–15$

Step 3: Combine like terms.

$x^2+4x+4=3x^2–4x–15$

Step 4: Move all terms to one side.

$0=2x^2–8x–19$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-8)\pm \sqrt{(-8{)}^2–4(2)(-19)}}{2(2)}$

$x=\frac{8\pm \sqrt{64+152}}{4}$

$x=\frac{8\pm \sqrt{216}}{4}$

$x=\frac{8\pm 14.7}{4}$

Step 6: Solve for $x$.

$x=\frac{8+14.7}{4}=5.175$

$x=\frac{8–14.7}{4}=-1.675$

Thus, the solutions are approximately $x=5.175$ and $x=-1.675$.

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Example 30:

Simplify the rational expression:

$\frac{6x^2–18}{x^2–9}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{6(x^2–3)}{(x–3)(x+3)}$

Step 2: Simplify.

$=\frac{6}{x+3}$

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Example 31:

Solve the rational equation:

$\frac{2x}{x+3}=\frac{5x+6}{x–2}$

Solution:

Step 1: Cross multiply.

$2x(x–2)=(5x+6)(x+3)$

Step 2: Expand both sides.

$2x^2–4x=5x^2+15x+6x+18$

Step 3: Combine like terms.

$2x^2–4x=5x^2+21x+18$

Step 4: Move all terms to one side.

$2x^2–4x–5x^2–21x–18=0$

$-3x^2–25x–18=0$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-25)\pm \sqrt{(-25{)}^2–4(-3)(-18)}}{2(-3)}$

$x=\frac{25\pm \sqrt{625–216}}{-6}$

$x=\frac{25\pm \sqrt{409}}{-6}$

Step 6: Solve for $x$.

$x=\frac{25+\sqrt{409}}{-6}$ or $x=\frac{25–\sqrt{409}}{-6}$

Thus, the solutions are complex and irrational.

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Example 32:

Simplify the rational expression:

$\frac{4x^2–9}{2x^2+5x–3}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(2x–3)(2x+3)}{(2x–3)(x+1)}$

Step 2: Cancel the common factor $2x–3$.

$=\frac{2x+3}{x+1}$

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Example 33:

Simplify the rational expression:

$\frac{5x^2+15x}{10x^2–30x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{5x(x+3)}{10x(x–3)}$

Step 2: Cancel the common factor $5x$.

$=\frac{x+3}{2(x–3)}$

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Example 34:

Add the rational expressions:

$\frac{3}{x–4}+\frac{5}{x+6}$

Solution:

Step 1: Find the LCD, which is $(x–4)(x+6)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{3(x+6)}{(x–4)(x+6)}+\frac{5(x–4)}{(x–4)(x+6)}$

Step 3: Combine the numerators.

$=\frac{3(x+6)+5(x–4)}{(x–4)(x+6)}$

Step 4: Expand the numerators.

$=\frac{3x+18+5x–20}{(x–4)(x+6)}$

Step 5: Combine like terms.

$=\frac{8x–2}{(x–4)(x+6)}$

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Example 35:

Subtract the rational expressions:

$\frac{7}{x+1}–\frac{2}{x+3}$

Solution:

Step 1: Find the LCD, which is $(x+1)(x+3)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{7(x+3)}{(x+1)(x+3)}–\frac{2(x+1)}{(x+1)(x+3)}$

Step 3: Combine the numerators.

$=\frac{7(x+3)–2(x+1)}{(x+1)(x+3)}$

Step 4: Expand the numerators.

$=\frac{7x+21–2x–2}{(x+1)(x+3)}$

Step 5: Combine like terms.

$=\frac{5x+19}{(x+1)(x+3)}$

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Example 36:

Multiply the rational expressions:

$\frac{4x}{x^2–16}\times \frac{x+4}{2x}$

Solution:

Step 1: Factor the denominator $x^2–16=(x–4)(x+4)$.

$=\frac{4x}{(x–4)(x+4)}\times \frac{x+4}{2x}$

Step 2: Cancel the common factor $x+4$ and $x$.

$=\frac{4}{2(x–4)}$

Step 3: Simplify.

$=\frac{2}{x–4}$

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Example 37:

Divide the rational expressions:

$\frac{x^2+3x}{x^2–9}÷\frac{2x+6}{x–3}$

Solution:

Step 1: Rewrite the division as multiplication by the reciprocal.

$=\frac{x(x+3)}{(x–3)(x+3)}\times \frac{x–3}{2(x+3)}$

Step 2: Cancel the common factor $x–3$ and $x+3$.

$=\frac{x}{2}$

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Example 38:

Solve the rational equation:

$\frac{x+4}{x–2}=\frac{2x+3}{x+1}$

Solution:

Step 1: Cross multiply.

$(x+4)(x+1)=(2x+3)(x–2)$

Step 2: Expand both sides.

$x^2+x+4x+4=2x^2–4x+3x–6$

Step 3: Combine like terms.

$x^2+5x+4=2x^2–x–6$

Step 4: Move all terms to one side.

$0=x^2–6x–10$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-6)\pm \sqrt{(-6{)}^2–4(1)(-10)}}{2(1)}$

$x=\frac{6\pm \sqrt{36+40}}{2}$

$x=\frac{6\pm \sqrt{76}}{2}$

$x=\frac{6\pm 8.72}{2}$

Step 6: Solve for $x$.

$x=\frac{6+8.72}{2}=7.36$

$x=\frac{6–8.72}{2}=-1.36$

Thus, the solutions are $x=7.36$ and $x=-1.36$.

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Example 39:

Simplify the rational expression:

$\frac{9x^2–36}{x^2+2x–24}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{9(x^2–4)}{(x–4)(x+6)}$

Step 2: Factor $x^2–4=(x–2)(x+2)$.

$=\frac{9(x–2)(x+2)}{(x–4)(x+6)}$

No common factors exist, so the simplified form is:

$=\frac{9(x–2)(x+2)}{(x–4)(x+6)}$

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Example 40:

Add the rational expressions:

$\frac{2x}{x–1}+\frac{3}{x+2}$

Solution:

Step 1: Find the LCD, which is $(x–1)(x+2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{2x(x+2)}{(x–1)(x+2)}+\frac{3(x–1)}{(x–1)(x+2)}$

Step 3: Combine the numerators.

$=\frac{2x(x+2)+3(x–1)}{(x–1)(x+2)}$

Step 4: Expand the numerators.

$=\frac{2x^2+4x+3x–3}{(x–1)(x+2)}$

Step 5: Combine like terms.

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Example 41:

Subtract the rational expressions:

$\frac{4x}{x–3}–\frac{5}{x+2}$

Solution:

Step 1: Find the LCD, which is $(x–3)(x+2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{4x(x+2)}{(x–3)(x+2)}–\frac{5(x–3)}{(x–3)(x+2)}$

Step 3: Combine the numerators.

$=\frac{4x(x+2)–5(x–3)}{(x–3)(x+2)}$

Step 4: Expand the numerators.

$=\frac{4x^2+8x–5x+15}{(x–3)(x+2)}$

Step 5: Combine like terms.

$=\frac{4x^2+3x+15}{(x–3)(x+2)}$

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Example 42:

Multiply the rational expressions:

$\frac{2x+1}{x^2–1}\times \frac{x+1}{x–2}$

Solution:

Step 1: Factor the denominator $x^2–1=(x–1)(x+1)$.

$=\frac{2x+1}{(x–1)(x+1)}\times \frac{x+1}{x–2}$

Step 2: Multiply the numerators and the denominators.

$=\frac{(2x+1)(x+1)}{(x–1)(x+1)(x–2)}$

Step 3: Cancel the common factor $(x+1)$.

$=\frac{2x+1}{(x–1)(x–2)}$

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Example 43:

Divide the rational expressions:

$\frac{5x}{x^2–9}÷\frac{2x}{x+3}$

Solution:

Step 1: Rewrite the division as multiplication by the reciprocal.

$=\frac{5x}{(x–3)(x+3)}\times \frac{x+3}{2x}$

Step 2: Cancel the common factor $x+3$ and $x$.

$=\frac{5}{2(x–3)}$

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Example 44:

Solve the rational equation:

$\frac{x+1}{x–4}=\frac{2x+5}{x+2}$

Solution:

Step 1: Cross multiply.

$(x+1)(x+2)=(2x+5)(x–4)$

Step 2: Expand both sides.

$x^2+2x+x+2=2x^2–8x+5x–20$

Step 3: Combine like terms.

$x^2+3x+2=2x^2–3x–20$

Step 4: Move all terms to one side.

$0=x^2–6x–22$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-6)\pm \sqrt{(-6{)}^2–4(1)(-22)}}{2(1)}$

$x=\frac{6\pm \sqrt{36+88}}{2}$

$x=\frac{6\pm \sqrt{124}}{2}$

$x=\frac{6\pm 11.14}{2}$

Step 6: Solve for $x$.

$x=\frac{6+11.14}{2}=8.57$

$x=\frac{6–11.14}{2}=-2.57$

Thus, the solutions are approximately $x=8.57$ and $x=-2.57$.

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Example 45:

Simplify the rational expression:

$\frac{4x^2+12x}{2x^2+6x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{4x(x+3)}{2x(x+3)}$

Step 2: Cancel the common factor $x(x+3)$.

$=\frac{2}{1}$

Thus, the simplified form is $2$.

---

Example 46:

Simplify the rational expression:

$\frac{x^2–9}{x^2+2x–3}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–3)(x+3)}{(x+3)(x–1)}$

Step 2: Cancel the common factor $(x+3)$.

$=\frac{x–3}{x–1}$

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Example 47:

Add the rational expressions:

$\frac{x+5}{x–2}+\frac{3x}{x+1}$

Solution:

Step 1: Find the LCD, which is $(x–2)(x+1)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{(x+5)(x+1)}{(x–2)(x+1)}+\frac{3x(x–2)}{(x–2)(x+1)}$

Step 3: Combine the numerators.

$=\frac{(x+5)(x+1)+3x(x–2)}{(x–2)(x+1)}$

Step 4: Expand the numerators.

$=\frac{x^2+5x+x+5+3x^2–6x}{(x–2)(x+1)}$

Step 5: Combine like terms.

$=\frac{4x^2+x+5}{(x–2)(x+1)}$

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Example 48:

Subtract the rational expressions:

$\frac{2x+7}{x–3}–\frac{4}{x+1}$

Solution:

Step 1: Find the LCD, which is $(x–3)(x+1)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{(2x+7)(x+1)}{(x–3)(x+1)}–\frac{4(x–3)}{(x–3)(x+1)}$

Step 3: Combine the numerators.

$=\frac{(2x+7)(x+1)–4(x–3)}{(x–3)(x+1)}$

Step 4: Expand the numerators.

$=\frac{2x^2+2x+7x+7–4x+12}{(x–3)(x+1)}$

Step 5: Combine like terms.

$=\frac{2x^2+5x+19}{(x–3)(x+1)}$

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Example 49:

Multiply the rational expressions:

$\frac{x+2}{x^2–4}\times \frac{2x–4}{x+2}$

Solution:

Step 1: Factor the denominator $x^2–4=(x–2)(x+2)$.

$=\frac{x+2}{(x–2)(x+2)}\times \frac{2(x–2)}{x+2}$

Step 2: Cancel the common factor $x+2$ and $x–2$.

$=\frac{2}{1}$

Thus, the simplified form is $2$.

---

Example 50:

Divide the rational expressions:

$\frac{5x}{x^2+6x+9}÷\frac{x}{x+3}$

Solution:

Step 1: Rewrite the division as multiplication by the reciprocal.

$=\frac{5x}{(x+3{)}^2}\times \frac{x+3}{x}$

Step 2: Cancel the common factor $x$ and $x+3$.

$=\frac{5}{x+3}$

---

xample 51:

Solve the rational equation:

$\frac{x+4}{x–2}=\frac{2x+6}{x+3}$

Solution:

Step 1: Cross multiply.

$(x+4)(x+3)=(2x+6)(x–2)$

Step 2: Expand both sides.

$x^2+3x+4x+12=2x^2–4x+6x–12$

Step 3: Combine like terms.

$x^2+7x+12=2x^2+2x–12$

Step 4: Move all terms to one side.

$0=x^2–5x–24$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-5)\pm \sqrt{(-5{)}^2–4(1)(-24)}}{2(1)}$

$x=\frac{5\pm \sqrt{25+96}}{2}$

$x=\frac{5\pm \sqrt{121}}{2}$

$x=\frac{5\pm 11}{2}$

Step 6: Solve for $x$.

$x=\frac{5+11}{2}=8$

$x=\frac{5–11}{2}=-3$

Thus, the solutions are $x=8$ and $x=-3$.

---

Example 52:

Simplify the rational expression:

$\frac{4x^2–25}{2x^2–10x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(2x–5)(2x+5)}{2x(x–5)}$

Step 2: Cancel the common factor $(2x–5)$.

$=\frac{2x+5}{2x}$

---

Example 53:

Simplify the rational expression:

$\frac{6x^2+9x}{3x^2–15x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{3x(2x+3)}{3x(x–5)}$

Step 2: Cancel the common factor $3x$.

$=\frac{2x+3}{x–5}$

---

Example 54:

Add the rational expressions:

$\frac{7}{x–4}+\frac{5x}{x+6}$

Solution:

Step 1: Find the LCD, which is $(x–4)(x+6)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{7(x+6)}{(x–4)(x+6)}+\frac{5x(x–4)}{(x–4)(x+6)}$

Step 3: Combine the numerators.

$=\frac{7(x+6)+5x(x–4)}{(x–4)(x+6)}$

Step 4: Expand the numerators.

$=\frac{7x+42+5x^2–20x}{(x–4)(x+6)}$

Step 5: Combine like terms.

$=\frac{5x^2–13x+42}{(x–4)(x+6)}$

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Example 55:

Subtract the rational expressions:

$\frac{9}{x–5}–\frac{4x}{x+2}$

Solution:

Step 1: Find the LCD, which is $(x–5)(x+2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{9(x+2)}{(x–5)(x+2)}–\frac{4x(x–5)}{(x–5)(x+2)}$

Step 3: Combine the numerators.

$=\frac{9(x+2)–4x(x–5)}{(x–5)(x+2)}$

Step 4: Expand the numerators.

$=\frac{9x+18–4x^2+20x}{(x–5)(x+2)}$

Step 5: Combine like terms.

$=\frac{-4x^2+29x+18}{(x–5)(x+2)}$

---

Example 56:

Multiply the rational expressions:

$\frac{5x}{x^2–9}\times \frac{x+3}{2x}$

Solution:

Step 1: Factor the denominator $x^2–9=(x–3)(x+3)$.

$=\frac{5x}{(x–3)(x+3)}\times \frac{x+3}{2x}$

Step 2: Cancel the common factor $x+3$ and $x$.

$=\frac{5}{2(x–3)}$

---

Example 57:

Divide the rational expressions:

$\frac{x+2}{x^2–4x+4}÷\frac{2x}{x–2}$

Solution:

Step 1: Factor the denominator $x^2–4x+4=(x–2{)}^2$.

$=\frac{x+2}{(x–2{)}^2}÷\frac{2x}{x–2}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{x+2}{(x–2{)}^2}\times \frac{x–2}{2x}$

Step 3: Cancel the common factor $x–2$.

$=\frac{x+2}{2x(x–2)}$

---

Example 58:

Solve the rational equation:

$\frac{2x+3}{x–1}=\frac{5x–2}{x+4}$

Solution:

Step 1: Cross multiply.

$(2x+3)(x+4)=(5x–2)(x–1)$

Step 2: Expand both sides.

$2x^2+8x+3x+12=5x^2–5x–2x+2$

Step 3: Combine like terms.

$2x^2+11x+12=5x^2–7x+2$

Step 4: Move all terms to one side.

$0=3x^2–18x–10$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-18)\pm \sqrt{(-18{)}^2–4(3)(-10)}}{2(3)}$

$x=\frac{18\pm \sqrt{324+120}}{6}$

$x=\frac{18\pm \sqrt{444}}{6}$

$x=\frac{18\pm 21.07}{6}$

Step 6: Solve for $x$.

$x=\frac{18+21.07}{6}=6.51$

$x=\frac{18–21.07}{6}=-0.51$

Thus, the solutions are $x=6.51$ and $x=-0.51$.

---

Example 59:

Simplify the rational expression:

$\frac{6x^2+11x+3}{3x^2–3x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(2x+3)(3x+1)}{3x(x–1)}$

No common factors exist, so the simplified form is:

$=\frac{(2x+3)(3x+1)}{3x(x–1)}$

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Example 60:

Add the rational expressions:

$\frac{3}{x–2}+\frac{4x}{x+5}$

Solution:

Step 1: Find the LCD, which is $(x–2)(x+5)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{3(x+5)}{(x–2)(x+5)}+\frac{4x(x–2)}{(x–2)(x+5)}$

Step 3: Combine the numerators.

$=\frac{3(x+5)+4x(x–2)}{(x–2)(x+5)}$

Step 4: Expand the numerators.

$=\frac{3x+15+4x^2–8x}{(x–2)(x+5)}$

Step 5: Combine like terms.

$=\frac{4x^2–5x+15}{(x–2)(x+5)}$

---

Example 61:

Subtract the rational expressions:

$\frac{5x}{x–3}–\frac{6}{x+4}$

Solution:

Step 1: Find the LCD, which is $(x–3)(x+4)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{5x(x+4)}{(x–3)(x+4)}–\frac{6(x–3)}{(x–3)(x+4)}$

Step 3: Combine the numerators.

$=\frac{5x(x+4)–6(x–3)}{(x–3)(x+4)}$

Step 4: Expand the numerators.

$=\frac{5x^2+20x–6x+18}{(x–3)(x+4)}$

Step 5: Combine like terms.

$=\frac{5x^2+14x+18}{(x–3)(x+4)}$

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Example 62:

Multiply the rational expressions:

$\frac{x^2–9}{x+3}\times \frac{x–3}{x^2+2x+1}$

Solution:

Step 1: Factor the numerator and denominator wherever possible.

$=\frac{(x–3)(x+3)}{x+3}\times \frac{x–3}{(x+1{)}^2}$

Step 2: Cancel the common factors $(x+3)$ and $(x–3)$.

$=\frac{1}{(x+1{)}^2}$

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Example 63:

Divide the rational expressions:

$\frac{x+2}{x^2–9}÷\frac{3x+9}{x–3}$

Solution:

Step 1: Factor the denominator $x^2–9=(x–3)(x+3)$ and factor the numerator of the second expression.

$=\frac{x+2}{(x–3)(x+3)}÷\frac{3(x+3)}{x–3}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{x+2}{(x–3)(x+3)}\times \frac{x–3}{3(x+3)}$

Step 3: Cancel the common factors $x–3$ and $x+3$.

$=\frac{x+2}{3}$

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Example 64:

Solve the rational equation:

$\frac{3x+1}{x–2}=\frac{2x–3}{x+4}$

Solution:

Step 1: Cross multiply.

$(3x+1)(x+4)=(2x–3)(x–2)$

Step 2: Expand both sides.

$3x^2+12x+x+4=2x^2–4x–3x+6$

Step 3: Combine like terms.

$3x^2+13x+4=2x^2–7x+6$

Step 4: Move all terms to one side.

$x^2+20x–2=0$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-20\pm \sqrt{(20{)}^2–4(1)(-2)}}{2(1)}$

$x=\frac{-20\pm \sqrt{400+8}}{2}$

$x=\frac{-20\pm \sqrt{408}}{2}$

$x=\frac{-20\pm 20.19}{2}$

Step 6: Solve for $x$.

$x=\frac{-20+20.19}{2}=0.095$

$x=\frac{-20–20.19}{2}=-20.095$

Thus, the solutions are approximately $x=0.095$ and $x=-20.095$.

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Example 65:

Simplify the rational expression:

$\frac{4x^2–9x+2}{2x^2+5x–3}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(4x–1)(x–2)}{(2x+3)(x–1)}$

No common factors exist, so the simplified form is:

$=\frac{(4x–1)(x–2)}{(2x+3)(x–1)}$

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Example 66:

Add the rational expressions:

$\frac{2x}{x–3}+\frac{4}{x+5}$

Solution:

Step 1: Find the LCD, which is $(x–3)(x+5)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{2x(x+5)}{(x–3)(x+5)}+\frac{4(x–3)}{(x–3)(x+5)}$

Step 3: Combine the numerators.

$=\frac{2x(x+5)+4(x–3)}{(x–3)(x+5)}$

Step 4: Expand the numerators.

$=\frac{2x^2+10x+4x–12}{(x–3)(x+5)}$

Step 5: Combine like terms.

$=\frac{2x^2+14x–12}{(x–3)(x+5)}$

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Example 67:

Subtract the rational expressions:

$\frac{5x}{x–2}–\frac{7}{x+3}$

Solution:

Step 1: Find the LCD, which is $(x–2)(x+3)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{5x(x+3)}{(x–2)(x+3)}–\frac{7(x–2)}{(x–2)(x+3)}$

Step 3: Combine the numerators.

$=\frac{5x(x+3)–7(x–2)}{(x–2)(x+3)}$

Step 4: Expand the numerators.

$=\frac{5x^2+15x–7x+14}{(x–2)(x+3)}$

Step 5: Combine like terms.

$=\frac{5x^2+8x+14}{(x–2)(x+3)}$

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Example 68:

Multiply the rational expressions:

$\frac{x^2–4x}{x^2–9}\times \frac{x+3}{x–2}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{x(x–4)}{(x–3)(x+3)}\times \frac{x+3}{x–2}$

Step 2: Cancel the common factor $x+3$.

$=\frac{x(x–4)}{(x–3)(x–2)}$

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Example 69:

Divide the rational expressions:

$\frac{x^2–x–6}{x+2}÷\frac{x^2–9}{x+3}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–3)(x+2)}{x+2}÷\frac{(x–3)(x+3)}{x+3}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{(x–3)(x+2)}{x+2}\times \frac{x+3}{(x–3)(x+3)}$

Step 3: Cancel the common factors.

$=1$

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Example 70:

Solve the rational equation:

$\frac{2x+1}{x+4}=\frac{5x–2}{x–3}$

Solution:

Step 1: Cross multiply.

$(2x+1)(x–3)=(5x–2)(x+4)$

Step 2: Expand both sides.

$2x^2–6x+x–3=5x^2+20x–2x–8$

Step 3: Combine like terms.

$2x^2–5x–3=5x^2+18x–8$

Step 4: Move all terms to one side.

$0=3x^2+23x+11$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-23\pm \sqrt{(23{)}^2–4(3)(11)}}{2(3)}$

$x=\frac{-23\pm \sqrt{529–132}}{6}$

$x=\frac{-23\pm \sqrt{397}}{6}$

Thus, the solutions are complex.

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Example 71:

Simplify the rational expression:

$\frac{3x^2–12x}{6x^2–18x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{3x(x–4)}{6x(x–3)}$

Step 2: Cancel the common factor $3x$.

$=\frac{x–4}{2(x–3)}$

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Example 72:

Simplify the rational expression:

$\frac{x^2–9x+14}{x^2–7x+12}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–7)(x–2)}{(x–3)(x–4)}$

No common factors exist, so the simplified form is:

$=\frac{(x–7)(x–2)}{(x–3)(x–4)}$

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Example 73:

Add the rational expressions:

$\frac{4}{x–1}+\frac{7}{x+5}$

Solution:

Step 1: Find the LCD, which is $(x–1)(x+5)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{4(x+5)}{(x–1)(x+5)}+\frac{7(x–1)}{(x–1)(x+5)}$

Step 3: Combine the numerators.

$=\frac{4(x+5)+7(x–1)}{(x–1)(x+5)}$

Step 4: Expand the numerators.

$=\frac{4x+20+7x–7}{(x–1)(x+5)}$

Step 5: Combine like terms.

$=\frac{11x+13}{(x–1)(x+5)}$

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Example 74:

Subtract the rational expressions:

$\frac{6x}{x–2}–\frac{5}{x+3}$

Solution:

Step 1: Find the LCD, which is $(x–2)(x+3)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{6x(x+3)}{(x–2)(x+3)}–\frac{5(x–2)}{(x–2)(x+3)}$

Step 3: Combine the numerators.

$=\frac{6x(x+3)–5(x–2)}{(x–2)(x+3)}$

Step 4: Expand the numerators.

$=\frac{6x^2+18x–5x+10}{(x–2)(x+3)}$

Step 5: Combine like terms.

$=\frac{6x^2+13x+10}{(x–2)(x+3)}$

---

Example 75:

Multiply the rational expressions:

$\frac{x^2+6x}{x–1}\times \frac{2x–2}{x+6}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{x(x+6)}{x–1}\times \frac{2(x–1)}{x+6}$

Step 2: Cancel the common factors $x+6$ and $x–1$.

$=\frac{2x}{1}$

Thus, the simplified form is $2x$.

---

Example 76:

Divide the rational expressions:

$\frac{3x+6}{x^2–4}÷\frac{5x}{x+2}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{3(x+2)}{(x–2)(x+2)}÷\frac{5x}{x+2}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{3(x+2)}{(x–2)(x+2)}\times \frac{x+2}{5x}$

Step 3: Cancel the common factors $x+2$.

$=\frac{3}{5x(x–2)}$

---

Example 77:

Solve the rational equation:

$\frac{x+1}{x–3}=\frac{2x–1}{x+4}$

Solution:

Step 1: Cross multiply.

$(x+1)(x+4)=(2x–1)(x–3)$

Step 2: Expand both sides.

$x^2+4x+x+4=2x^2–6x–x+3$

Step 3: Combine like terms.

$x^2+5x+4=2x^2–7x+3$

Step 4: Move all terms to one side.

$0=x^2–12x–1$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-12)\pm \sqrt{(-12{)}^2–4(1)(-1)}}{2(1)}$

$x=\frac{12\pm \sqrt{144+4}}{2}$

$x=\frac{12\pm \sqrt{148}}{2}$

$x=\frac{12\pm 12.17}{2}$

Step 6: Solve for $x$.

$x=\frac{12+12.17}{2}=12.08$

$x=\frac{12–12.17}{2}=-0.085$

Thus, the solutions are approximately $x=12.08$ and $x=-0.085$.

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Example 78:

Simplify the rational expression:

$\frac{x^2–25}{x^2–x–20}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–5)(x+5)}{(x–5)(x+4)}$

Step 2: Cancel the common factor $x–5$.

$=\frac{x+5}{x+4}$

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Example 79:

Add the rational expressions:

$\frac{x+4}{x–1}+\frac{3x}{x+6}$

Solution:

Step 1: Find the LCD, which is $(x–1)(x+6)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{(x+4)(x+6)}{(x–1)(x+6)}+\frac{3x(x–1)}{(x–1)(x+6)}$

Step 3: Combine the numerators.

$=\frac{(x+4)(x+6)+3x(x–1)}{(x–1)(x+6)}$

Step 4: Expand the numerators.

$=\frac{x^2+6x+4x+24+3x^2–3x}{(x–1)(x+6)}$

Step 5: Combine like terms.

$=\frac{4x^2+7x+24}{(x–1)(x+6)}$

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Example 80:

Subtract the rational expressions:

$\frac{5x+2}{x–4}–\frac{3x}{x+5}$

Solution:

Step 1: Find the LCD, which is $(x–4)(x+5)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{(5x+2)(x+5)}{(x–4)(x+5)}–\frac{3x(x–4)}{(x–4)(x+5)}$

Step 3: Combine the numerators.

$=\frac{(5x+2)(x+5)–3x(x–4)}{(x–4)(x+5)}$

Step 4: Expand the numerators.

$=\frac{5x^2+25x+2x+10–3x^2+12x}{(x–4)(x+5)}$

Step 5: Combine like terms.

$=\frac{2x^2+39x+10}{(x–4)(x+5)}$

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Example 81:

Multiply the rational expressions:

$\frac{x^2–4x}{x+2}\times \frac{x+3}{x–2}$

Solution:

Step 1: Factor both the numerator and the denominator if possible.

$=\frac{x(x–4)}{x+2}\times \frac{x+3}{x–2}$

Step 2: Since there are no common factors to cancel, multiply the numerators and the denominators.

$=\frac{x(x–4)(x+3)}{(x+2)(x–2)}$

Thus, the simplified form is:

$=\frac{x(x–4)(x+3)}{(x+2)(x–2)}$

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Example 82:

Divide the rational expressions:

$\frac{x^2–9}{x^2+3x}÷\frac{2x}{x+3}$

Solution:

Step 1: Factor the numerator and the denominator.

$=\frac{(x–3)(x+3)}{x(x+3)}÷\frac{2x}{x+3}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{(x–3)(x+3)}{x(x+3)}\times \frac{x+3}{2x}$

Step 3: Cancel the common factors $x+3$ and $x$.

$=\frac{x–3}{2}$

Thus, the simplified form is $\frac{x–3}{2}$.

---

Example 83:

Solve the rational equation:

$\frac{x+1}{x–3}=\frac{2x–5}{x+4}$

Solution:

Step 1: Cross multiply.

$(x+1)(x+4)=(2x–5)(x–3)$

Step 2: Expand both sides.

$x^2+4x+x+4=2x^2–6x–5x+15$

Step 3: Combine like terms.

$x^2+5x+4=2x^2–11x+15$

Step 4: Move all terms to one side.

$0=x^2–16x+11$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-16)\pm \sqrt{(-16{)}^2–4(1)(11)}}{2(1)}$

$x=\frac{16\pm \sqrt{256–44}}{2}$

$x=\frac{16\pm \sqrt{212}}{2}$

$x=\frac{16\pm 14.56}{2}$

Step 6: Solve for $x$.

$x=\frac{16+14.56}{2}=15.28$

$x=\frac{16–14.56}{2}=0.72$

Thus, the solutions are approximately $x=15.28$ and $x=0.72$.

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Example 84:

Simplify the rational expression:

$\frac{2x^2+7x+3}{x^2+5x+6}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(2x+3)(x+1)}{(x+2)(x+3)}$

Step 2: Cancel the common factor $x+3$.

$=\frac{2x+1}{x+2}$

Thus, the simplified form is $\frac{2x+1}{x+2}$.

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Example 85:

Add the rational expressions:

$\frac{2}{x+3}+\frac{4}{x–2}$

Solution:

Step 1: Find the LCD, which is $(x+3)(x–2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{2(x–2)}{(x+3)(x–2)}+\frac{4(x+3)}{(x+3)(x–2)}$

Step 3: Combine the numerators.

$=\frac{2(x–2)+4(x+3)}{(x+3)(x–2)}$

Step 4: Expand the numerators.

$=\frac{2x–4+4x+12}{(x+3)(x–2)}$

Step 5: Combine like terms.

$=\frac{6x+8}{(x+3)(x–2)}$

Thus, the simplified form is $\frac{6x+8}{(x+3)(x–2)}$.

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Example 86:

Subtract the rational expressions:

$\frac{7x}{x+1}–\frac{3}{x–4}$

Solution:

Step 1: Find the LCD, which is $(x+1)(x–4)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{7x(x–4)}{(x+1)(x–4)}–\frac{3(x+1)}{(x+1)(x–4)}$

Step 3: Combine the numerators.

$=\frac{7x(x–4)–3(x+1)}{(x+1)(x–4)}$

Step 4: Expand the numerators.

$=\frac{7x^2–28x–3x–3}{(x+1)(x–4)}$

Step 5: Combine like terms.

$=\frac{7x^2–31x–3}{(x+1)(x–4)}$

Thus, the simplified form is $\frac{7x^2–31x–3}{(x+1)(x–4)}$.

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Example 87:

Multiply the rational expressions:

$\frac{x^2+2x–8}{x^2–9}\times \frac{x–3}{x+4}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x+4)(x–2)}{(x–3)(x+3)}\times \frac{x–3}{x+4}$

Step 2: Cancel the common factors $x–3$ and $x+4$.

$=\frac{x–2}{x+3}$

Thus, the simplified form is $\frac{x–2}{x+3}$.

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Example 88:

Divide the rational expressions:

$\frac{2x^2–7x+3}{x^2–4x+4}÷\frac{x–2}{x^2+x–6}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(2x–1)(x–3)}{(x–2{)}^2}÷\frac{x–2}{(x–2)(x+3)}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{(2x–1)(x–3)}{(x–2{)}^2}\times \frac{(x–2)(x+3)}{x–2}$

Step 3: Cancel the common factors $x–2$.

$=\frac{(2x–1)(x–3)(x+3)}{(x–2)}$

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Example 89:

Solve the rational equation:

$\frac{3x+4}{x–2}=\frac{2x+5}{x+3}$

Solution:

Step 1: Cross multiply.

$(3x+4)(x+3)=(2x+5)(x–2)$

Step 2: Expand both sides.

$3x^2+9x+4x+12=2x^2–4x+5x–10$

Step 3: Combine like terms.

$3x^2+13x+12=2x^2+x–10$

Step 4: Move all terms to one side.

$x^2+12x+22=0$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-12\pm \sqrt{{12}^2–4(1)(22)}}{2(1)}$

$x=\frac{-12\pm \sqrt{144–88}}{2}$

$x=\frac{-12\pm \sqrt{56}}{2}$

$x=\frac{-12\pm 7.48}{2}$

Step 6: Solve for $x$.

$x=\frac{-12+7.48}{2}=-2.26$

$x=\frac{-12–7.48}{2}=-9.74$

Thus, the solutions are approximately $x=-2.26$ and $x=-9.74$.

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Example 90:

Simplify the rational expression:

$\frac{4x^2+8x+4}{2x^2–4}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{4(x^2+2x+1)}{2(x^2–2)}$

Step 2: Factor further.

$=\frac{4(x+1{)}^2}{2(x–2)(x+2)}$

Step 3: Cancel common factors.

$=\frac{2(x+1{)}^2}{(x–2)(x+2)}$

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Example 91:

Add the rational expressions:

$\frac{3x}{x–1}+\frac{4}{x+2}$

Solution:

Step 1: Find the LCD, which is $(x–1)(x+2)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{3x(x+2)}{(x–1)(x+2)}+\frac{4(x–1)}{(x–1)(x+2)}$

Step 3: Combine the numerators.

$=\frac{3x(x+2)+4(x–1)}{(x–1)(x+2)}$

Step 4: Expand the numerators.

$=\frac{3x^2+6x+4x–4}{(x–1)(x+2)}$

Step 5: Combine like terms.

$=\frac{3x^2+10x–4}{(x–1)(x+2)}$

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Example 92:

Subtract the rational expressions:

$\frac{5x}{x–3}–\frac{2}{x+4}$

Solution:

Step 1: Find the LCD, which is $(x–3)(x+4)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{5x(x+4)}{(x–3)(x+4)}–\frac{2(x–3)}{(x–3)(x+4)}$

Step 3: Combine the numerators.

$=\frac{5x(x+4)–2(x–3)}{(x–3)(x+4)}$

Step 4: Expand the numerators.

$=\frac{5x^2+20x–2x+6}{(x–3)(x+4)}$

Step 5: Combine like terms.

$=\frac{5x^2+18x+6}{(x–3)(x+4)}$

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Example 93:

Multiply the rational expressions:

$\frac{x^2–4}{x+3}\times \frac{x–2}{x+4}$

Solution:

Step 1: Factor the numerator and denominator if possible.

$=\frac{(x–2)(x+2)}{x+3}\times \frac{x–2}{x+4}$

Step 2: Cancel the common factor $x–2$.

$=\frac{x+2}{(x+3)(x+4)}$

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Example 94:

Divide the rational expressions:

$\frac{2x+1}{x^2–9}÷\frac{3x}{x–3}$

Solution:

Step 1: Factor the denominator $x^2–9=(x–3)(x+3)$.

$=\frac{2x+1}{(x–3)(x+3)}÷\frac{3x}{x–3}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{2x+1}{(x–3)(x+3)}\times \frac{x–3}{3x}$

Step 3: Cancel the common factor $x–3$.

$=\frac{2x+1}{3x(x+3)}$

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Example 95:

Solve the rational equation:

$\frac{x+5}{x–2}=\frac{3x–1}{x+4}$

Solution:

Step 1: Cross multiply.

$(x+5)(x+4)=(3x–1)(x–2)$

Step 2: Expand both sides.

$x^2+4x+5x+20=3x^2–6x–x+2$

Step 3: Combine like terms.

$x^2+9x+20=3x^2–7x+2$

Step 4: Move all terms to one side.

$0=2x^2–16x–18$

Step 5: Solve the quadratic equation using the quadratic formula.

$x=\frac{-(-16)\pm \sqrt{(-16{)}^2–4(2)(-18)}}{2(2)}$

$x=\frac{16\pm \sqrt{256+144}}{4}$

$x=\frac{16\pm \sqrt{400}}{4}$

$x=\frac{16\pm 20}{4}$

Step 6: Solve for $x$.

$x=\frac{16+20}{4}=9$

$x=\frac{16–20}{4}=-1$

Thus, the solutions are $x=9$ and $x=-1$.

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Example 96:

Simplify the rational expression:

$\frac{4x^2–16}{2x^2–8x}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{4(x^2–4)}{2x(x–4)}$

Step 2: Factor further.

$=\frac{4(x–2)(x+2)}{2x(x–4)}$

Step 3: Simplify by canceling common factors.

$=\frac{2(x–2)(x+2)}{x(x–4)}$

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Example 97:

Add the rational expressions:

$\frac{3}{x–2}+\frac{5}{x+4}$

Solution:

Step 1: Find the LCD, which is $(x–2)(x+4)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{3(x+4)}{(x–2)(x+4)}+\frac{5(x–2)}{(x–2)(x+4)}$

Step 3: Combine the numerators.

$=\frac{3(x+4)+5(x–2)}{(x–2)(x+4)}$

Step 4: Expand the numerators.

$=\frac{3x+12+5x–10}{(x–2)(x+4)}$

Step 5: Combine like terms.

$=\frac{8x+2}{(x–2)(x+4)}$

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Example 98:

Subtract the rational expressions:

$\frac{6x}{x+3}–\frac{4}{x–5}$

Solution:

Step 1: Find the LCD, which is $(x+3)(x–5)$.

Step 2: Rewrite each expression with the LCD.

$=\frac{6x(x–5)}{(x+3)(x–5)}–\frac{4(x+3)}{(x+3)(x–5)}$

Step 3: Combine the numerators.

$=\frac{6x(x–5)–4(x+3)}{(x+3)(x–5)}$

Step 4: Expand the numerators.

$=\frac{6x^2–30x–4x–12}{(x+3)(x–5)}$

Step 5: Combine like terms.

$=\frac{6x^2–34x–12}{(x+3)(x–5)}$

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Example 99:

Multiply the rational expressions:

$\frac{x^2–16}{x^2+4x+4}\times \frac{x+2}{x–4}$

Solution:

Step 1: Factor the numerator and denominator.

$=\frac{(x–4)(x+4)}{(x+2)(x+2)}\times \frac{x+2}{x–4}$

Step 2: Cancel the common factors $x–4$ and $x+2$.

$=\frac{x+4}{x+2}$

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Example 100:

Divide the rational expressions:

$\frac{x^2–9}{x^2–1}÷\frac{x–1}{x+3}$

Solution:

Step 1: Factor both the numerator and the denominator.

$=\frac{(x–3)(x+3)}{(x–1)(x+1)}÷\frac{x–1}{x+3}$

Step 2: Rewrite the division as multiplication by the reciprocal.

$=\frac{(x–3)(x+3)}{(x–1)(x+1)}\times \frac{x+3}{x–1}$

Step 3: Cancel the common factors $x–1$ and $x+3$.

$=\frac{x–3}{x+1}$