Absolute Value Functions: Detailed Explanation and 100 Solved Examples

Table of Contents

Introduction to Absolute Value Functions

An absolute value function is written as $f (x) = | x |$ , where $| x |$ represents the distance of $x$ from 0 on the number line. The absolute value of $x$ is always non-negative. It’s defined as:

$| x | = {\begin{cases} x, & if x \geq 0 - x, & if x < 0 \end{cases}$

Absolute value functions have a V-shaped graph, with the vertex at the origin if the function is $f (x) = | x |$ . If there are shifts or stretches, the vertex will change accordingly.

Some key properties:

$| a | \geq 0$ for any real number $a$ .
$| a | = 0$ if and only if $a = 0$ .
$| a b | = | a | | b |$ .
$| \frac{a}{b} | = \frac{| a |}{| b |}$ .
$| a + b | \leq | a | + | b |$ (Triangle Inequality).

Example 1: Solve $| x | = 5$

Solution:

Step 1: The absolute value function equals 5, so we split into two cases.
$x = 5$ or $x = - 5$

Thus, the solutions are $x = 5$ and $x = - 5$ .

Example 2: Solve $| x + 3 | = 7$

Solution:

Step 1: Split into two cases.

$x + 3 = 7$ or $x + 3 = - 7$

Step 2: Solve both cases.

$x = 7 - 3 = 4$
$x = - 7 - 3 = - 10$

Thus, the solutions are $x = 4$ and $x = - 10$ .

Example 3: Solve $| 2 x - 1 | = 9$

Solution:

Step 1: Split into two cases.

$2 x - 1 = 9$ or $2 x - 1 = - 9$

Step 2: Solve both cases.

For $2 x - 1 = 9$ :
$2 x = 9 + 1 = 10$
$x = \frac{10}{2} = 5$

For $2 x - 1 = - 9$ :
$2 x = - 9 + 1 = - 8$
$x = \frac{- 8}{2} = - 4$

Thus, the solutions are $x = 5$ and $x = - 4$ .

Example 4: Solve $| 3 x + 2 | = 0$

Solution:

Step 1: The absolute value of an expression equals zero when the expression inside is zero.
$3 x + 2 = 0$

Step 2: Solve for $x$ .
$3 x = - 2$
$x = \frac{- 2}{3}$

Thus, the solution is $x = \frac{- 2}{3}$ .

Example 5: Solve $| x - 4 | = 3$

Solution:

Step 1: Split into two cases.

$x - 4 = 3$ or $x - 4 = - 3$

Step 2: Solve both cases.

For $x - 4 = 3$ :
$x = 3 + 4 = 7$

For $x - 4 = - 3$ :
$x = - 3 + 4 = 1$

Thus, the solutions are $x = 7$ and $x = 1$ .

Example 6: Solve $| 2 x + 5 | = 10$

Solution:

Step 1: Split into two cases.

$2 x + 5 = 10$ or $2 x + 5 = - 10$

Step 2: Solve both cases.

For $2 x + 5 = 10$ :
$2 x = 10 - 5 = 5$
$x = \frac{5}{2} = 2.5$

For $2 x + 5 = - 10$ :
$2 x = - 10 - 5 = - 15$
$x = \frac{- 15}{2} = - 7.5$

Thus, the solutions are $x = 2.5$ and $x = - 7.5$ .

Example 7: Solve $| x + 6 | = 4$

Solution:

Step 1: Split into two cases.

$x + 6 = 4$ or $x + 6 = - 4$

Step 2: Solve both cases.

For $x + 6 = 4$ :
$x = 4 - 6 = - 2$

For $x + 6 = - 4$ :
$x = - 4 - 6 = - 10$

Thus, the solutions are $x = - 2$ and $x = - 10$ .

Example 8: Solve $| 4 x - 7 | = 12$

Solution:

Step 1: Split into two cases.

$4 x - 7 = 12$ or $4 x - 7 = - 12$

Step 2: Solve both cases.

For $4 x - 7 = 12$ :
$4 x = 12 + 7 = 19$
$x = \frac{19}{4} = 4.75$

For $4 x - 7 = - 12$ :
$4 x = - 12 + 7 = - 5$
$x = \frac{- 5}{4} = - 1.25$

Thus, the solutions are $x = 4.75$ and $x = - 1.25$ .

Example 9: Solve $| x - 5 | = 8$

Solution:

Step 1: Split into two cases.

$x - 5 = 8$ or $x - 5 = - 8$

Step 2: Solve both cases.

For $x - 5 = 8$ :
$x = 8 + 5 = 13$

For $x - 5 = - 8$ :
$x = - 8 + 5 = - 3$

Thus, the solutions are $x = 13$ and $x = - 3$ .

Example 10: Solve $| 3 x + 4 | = 7$

Solution:

Step 1: Split into two cases.

$3 x + 4 = 7$ or $3 x + 4 = - 7$

Step 2: Solve both cases.

For $3 x + 4 = 7$ :
$3 x = 7 - 4 = 3$
$x = \frac{3}{3} = 1$

For $3 x + 4 = - 7$ :
$3 x = - 7 - 4 = - 11$
$x = \frac{- 11}{3}$

Thus, the solutions are $x = 1$ and $x = \frac{- 11}{3}$ .

Example 11: Solve $| x + 9 | = 6$

Solution:

Step 1: Split into two cases.

$x + 9 = 6$ or $x + 9 = - 6$

Step 2: Solve both cases.

For $x + 9 = 6$ :
$x = 6 - 9 = - 3$

For $x + 9 = - 6$ :
$x = - 6 - 9 = - 15$

Thus, the solutions are $x = - 3$ and $x = - 15$ .

Example 12: Solve $| 2 x - 3 | = 5$

Solution:

Step 1: Split into two cases.

$2 x - 3 = 5$ or $2 x - 3 = - 5$

Step 2: Solve both cases.

For $2 x - 3 = 5$ :
$2 x = 5 + 3 = 8$
$x = \frac{8}{2} = 4$

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

Thus, the solutions are $x = 4$ and $x = - 1$ .

Example 13: Solve $| x - 7 | = 9$

Solution:

Step 1: Split into two cases.

$x - 7 = 9$ or $x - 7 = - 9$

Step 2: Solve both cases.

For $x - 7 = 9$ :
$x = 9 + 7 = 16$

For $x - 7 = - 9$ :
$x = - 9 + 7 = - 2$

Thus, the solutions are $x = 16$ and $x = - 2$ .

Example 14: Solve $| 2 x + 1 | = 11$

Solution:

Step 1: Split into two cases.

$2 x + 1 = 11$ or $2 x + 1 = - 11$

Step 2: Solve both cases.

For $2 x + 1 = 11$ :
$2 x = 11 - 1 = 10$
$x = \frac{10}{2} = 5$

For $2 x + 1 = - 11$ :
$2 x = - 11 - 1 = - 12$
$x = \frac{- 12}{2} = - 6$

Thus, the solutions are $x = 5$ and $x = - 6$ .

Example 15: Solve $| x - 2 | = 6$

Solution:

Step 1: Split into two cases.

$x - 2 = 6$ or $x - 2 = - 6$

Step 2: Solve both cases.

For $x - 2 = 6$ :
$x = 6 + 2 = 8$

For $x - 2 = - 6$ :
$x = - 6 + 2 = - 4$

Thus, the solutions are $x = 8$ and $x = - 4$ .

Example 16: Solve $| 5 x - 3 | = 13$

Solution:

Step 1: Split into two cases.

$5 x - 3 = 13$ or $5 x - 3 = - 13$

Step 2: Solve both cases.

For $5 x - 3 = 13$ :
$5 x = 13 + 3 = 16$
$x = \frac{16}{5} = 3.2$

For $5 x - 3 = - 13$ :
$5 x = - 13 + 3 = - 10$
$x = \frac{- 10}{5} = - 2$

Thus, the solutions are $x = 3.2$ and $x = - 2$ .

Example 17: Solve $| x + 5 | = 12$

Solution:

Step 1: Split into two cases.

$x + 5 = 12$ or $x + 5 = - 12$

Step 2: Solve both cases.

For $x + 5 = 12$ :
$x = 12 - 5 = 7$

For $x + 5 = - 12$ :
$x = - 12 - 5 = - 17$

Thus, the solutions are $x = 7$ and $x = - 17$ .

Example 18: Solve $| 4 x - 9 | = 7$

Solution:

Step 1: Split into two cases.

$4 x - 9 = 7$ or $4 x - 9 = - 7$

Step 2: Solve both cases.

For $4 x - 9 = 7$ :
$4 x = 7 + 9 = 16$
$x = \frac{16}{4} = 4$

For $4 x - 9 = - 7$ :
$4 x = - 7 + 9 = 2$
$x = \frac{2}{4} = 0.5$

Thus, the solutions are $x = 4$ and $x = 0.5$ .

Example 19: Solve $| 2 x - 4 | = 8$

Solution:

Step 1: Split into two cases.

$2 x - 4 = 8$ or $2 x - 4 = - 8$

Step 2: Solve both cases.

For $2 x - 4 = 8$ :
$2 x = 8 + 4 = 12$
$x = \frac{12}{2} = 6$

For $2 x - 4 = - 8$ :
$2 x = - 8 + 4 = - 4$
$x = \frac{- 4}{2} = - 2$

Thus, the solutions are $x = 6$ and $x = - 2$ .

Example 20: Solve $| 3 x + 1 | = 10$

Solution:

Step 1: Split into two cases.

$3 x + 1 = 10$ or $3 x + 1 = - 10$

Step 2: Solve both cases.

For $3 x + 1 = 10$ :
$3 x = 10 - 1 = 9$
$x = \frac{9}{3} = 3$

For $3 x + 1 = - 10$ :
$3 x = - 10 - 1 = - 11$
$x = \frac{- 11}{3}$

Thus, the solutions are $x = 3$ and $x = \frac{- 11}{3}$ .

Example 21: Solve $| 5 x + 3 | = 15$

Solution:

Step 1: Split into two cases.

$5 x + 3 = 15$ or $5 x + 3 = - 15$

Step 2: Solve both cases.

For $5 x + 3 = 15$ :
$5 x = 15 - 3 = 12$
$x = \frac{12}{5} = 2.4$

For $5 x + 3 = - 15$ :
$5 x = - 15 - 3 = - 18$
$x = \frac{- 18}{5} = - 3.6$

Thus, the solutions are $x = 2.4$ and $x = - 3.6$ .

Example 22: Solve $| x - 8 | = 4$

Solution:

Step 1: Split into two cases.

$x - 8 = 4$ or $x - 8 = - 4$

Step 2: Solve both cases.

For $x - 8 = 4$ :
$x = 4 + 8 = 12$

For $x - 8 = - 4$ :
$x = - 4 + 8 = 4$

Thus, the solutions are $x = 12$ and $x = 4$ .

Example 23: Solve $| x + 4 | = 5$

Solution:

Step 1: Split into two cases.

$x + 4 = 5$ or $x + 4 = - 5$

Step 2: Solve both cases.

For $x + 4 = 5$ :
$x = 5 - 4 = 1$

For $x + 4 = - 5$ :
$x = - 5 - 4 = - 9$

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

Example 24: Solve $| 2 x + 1 | = 9$

Solution:

Step 1: Split into two cases.

$2 x + 1 = 9$ or $2 x + 1 = - 9$

Step 2: Solve both cases.

For $2 x + 1 = 9$ :
$2 x = 9 - 1 = 8$
$x = \frac{8}{2} = 4$

For $2 x + 1 = - 9$ :
$2 x = - 9 - 1 = - 10$
$x = \frac{- 10}{2} = - 5$

Thus, the solutions are $x = 4$ and $x = - 5$ .

Example 25: Solve $| 3 x - 2 | = 6$

Solution:

Step 1: Split into two cases.

$3 x - 2 = 6$ or $3 x - 2 = - 6$

Step 2: Solve both cases.

For $3 x - 2 = 6$ :
$3 x = 6 + 2 = 8$
$x = \frac{8}{3}$

For $3 x - 2 = - 6$ :
$3 x = - 6 + 2 = - 4$
$x = \frac{- 4}{3}$

Thus, the solutions are $x = \frac{8}{3}$ and $x = \frac{- 4}{3}$ .

Example 26: Solve $| x - 1 | = 9$

Solution:

Step 1: Split into two cases.

$x - 1 = 9$ or $x - 1 = - 9$

Step 2: Solve both cases.

For $x - 1 = 9$ :
$x = 9 + 1 = 10$

For $x - 1 = - 9$ :
$x = - 9 + 1 = - 8$

Thus, the solutions are $x = 10$ and $x = - 8$ .

Example 27: Solve $| 4 x - 3 | = 2$

Solution:

Step 1: Split into two cases.

$4 x - 3 = 2$ or $4 x - 3 = - 2$

Step 2: Solve both cases.

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

For $4 x - 3 = - 2$ :
$4 x = - 2 + 3 = 1$
$x = \frac{1}{4}$

Thus, the solutions are $x = \frac{5}{4}$ and $x = \frac{1}{4}$ .

Example 28: Solve $| x + 6 | = 3$

Solution:

Step 1: Split into two cases.

$x + 6 = 3$ or $x + 6 = - 3$

Step 2: Solve both cases.

For $x + 6 = 3$ :
$x = 3 - 6 = - 3$

For $x + 6 = - 3$ :
$x = - 3 - 6 = - 9$

Thus, the solutions are $x = - 3$ and $x = - 9$ .

Example 29: Solve $| 2 x - 1 | = 7$

Solution:

Step 1: Split into two cases.

$2 x - 1 = 7$ or $2 x - 1 = - 7$

Step 2: Solve both cases.

For $2 x - 1 = 7$ :
$2 x = 7 + 1 = 8$
$x = \frac{8}{2} = 4$

For $2 x - 1 = - 7$ :
$2 x = - 7 + 1 = - 6$
$x = \frac{- 6}{2} = - 3$

Thus, the solutions are $x = 4$ and $x = - 3$ .

Example 30: Solve $| 5 x + 4 | = 14$

Solution:

Step 1: Split into two cases.

$5 x + 4 = 14$ or $5 x + 4 = - 14$

Step 2: Solve both cases.

For $5 x + 4 = 14$ :
$5 x = 14 - 4 = 10$
$x = \frac{10}{5} = 2$

For $5 x + 4 = - 14$ :
$5 x = - 14 - 4 = - 18$
$x = \frac{- 18}{5} = - 3.6$

Thus, the solutions are $x = 2$ and $x = - 3.6$ .

Example 31: Solve $| x + 3 | = 5$

Solution:

Step 1: Split into two cases.

$x + 3 = 5$ or $x + 3 = - 5$

Step 2: Solve both cases.

For $x + 3 = 5$ :
$x = 5 - 3 = 2$

For $x + 3 = - 5$ :
$x = - 5 - 3 = - 8$

Thus, the solutions are $x = 2$ and $x = - 8$ .

Example 32: Solve $| 4 x - 6 | = 10$

Solution:

Step 1: Split into two cases.

$4 x - 6 = 10$ or $4 x - 6 = - 10$

Step 2: Solve both cases.

For $4 x - 6 = 10$ :
$4 x = 10 + 6 = 16$
$x = \frac{16}{4} = 4$

For $4 x - 6 = - 10$ :
$4 x = - 10 + 6 = - 4$
$x = \frac{- 4}{4} = - 1$

Thus, the solutions are $x = 4$ and $x = - 1$ .

Example 33: Solve $| 3 x + 2 | = 9$

Solution:

Step 1: Split into two cases.

$3 x + 2 = 9$ or $3 x + 2 = - 9$

Step 2: Solve both cases.

For $3 x + 2 = 9$ :
$3 x = 9 - 2 = 7$
$x = \frac{7}{3}$

For $3 x + 2 = - 9$ :
$3 x = - 9 - 2 = - 11$
$x = \frac{- 11}{3}$

Thus, the solutions are $x = \frac{7}{3}$ and $x = \frac{- 11}{3}$ .

Example 34: Solve $| x - 10 | = 7$

Solution:

Step 1: Split into two cases.

$x - 10 = 7$ or $x - 10 = - 7$

Step 2: Solve both cases.

For $x - 10 = 7$ :
$x = 7 + 10 = 17$

For $x - 10 = - 7$ :
$x = - 7 + 10 = 3$

Thus, the solutions are $x = 17$ and $x = 3$ .

Example 35: Solve $| 2 x + 7 | = 5$

Solution:

Step 1: Split into two cases.

$2 x + 7 = 5$ or $2 x + 7 = - 5$

Step 2: Solve both cases.

For $2 x + 7 = 5$ :
$2 x = 5 - 7 = - 2$
$x = \frac{- 2}{2} = - 1$

For $2 x + 7 = - 5$ :
$2 x = - 5 - 7 = - 12$
$x = \frac{- 12}{2} = - 6$

Thus, the solutions are $x = - 1$ and $x = - 6$ .

Example 36: Solve $| 5 x - 2 | = 8$

Solution:

Step 1: Split into two cases.

$5 x - 2 = 8$ or $5 x - 2 = - 8$

Step 2: Solve both cases.

For $5 x - 2 = 8$ :
$5 x = 8 + 2 = 10$
$x = \frac{10}{5} = 2$

For $5 x - 2 = - 8$ :
$5 x = - 8 + 2 = - 6$
$x = \frac{- 6}{5}$

Thus, the solutions are $x = 2$ and $x = \frac{- 6}{5}$ .

Example 37: Solve $| x + 4 | = 9$

Solution:

Step 1: Split into two cases.

$x + 4 = 9$ or $x + 4 = - 9$

Step 2: Solve both cases.

For $x + 4 = 9$ :
$x = 9 - 4 = 5$

For $x + 4 = - 9$ :
$x = - 9 - 4 = - 13$

Thus, the solutions are $x = 5$ and $x = - 13$ .

Example 38: Solve $| 6 x - 5 | = 11$

Solution:

Step 1: Split into two cases.

$6 x - 5 = 11$ or $6 x - 5 = - 11$

Step 2: Solve both cases.

For $6 x - 5 = 11$ :
$6 x = 11 + 5 = 16$
$x = \frac{16}{6} = \frac{8}{3}$

For $6 x - 5 = - 11$ :
$6 x = - 11 + 5 = - 6$
$x = \frac{- 6}{6} = - 1$

Thus, the solutions are $x = \frac{8}{3}$ and $x = - 1$ .

Example 39: Solve $| x - 3 | = 10$

Solution:

Step 1: Split into two cases.

$x - 3 = 10$ or $x - 3 = - 10$

Step 2: Solve both cases.

For $x - 3 = 10$ :
$x = 10 + 3 = 13$

For $x - 3 = - 10$ :
$x = - 10 + 3 = - 7$

Thus, the solutions are $x = 13$ and $x = - 7$ .

Example 40: Solve $| 7 x + 2 | = 16$

Solution:

Step 1: Split into two cases.

$7 x + 2 = 16$ or $7 x + 2 = - 16$

Step 2: Solve both cases.

For $7 x + 2 = 16$ :
$7 x = 16 - 2 = 14$
$x = \frac{14}{7} = 2$

For $7 x + 2 = - 16$ :
$7 x = - 16 - 2 = - 18$
$x = \frac{- 18}{7}$

Thus, the solutions are $x = 2$ and $x = \frac{- 18}{7}$ .

Example 41: Solve $| 3 x + 6 | = 12$

Solution:

Step 1: Split into two cases.

$3 x + 6 = 12$ or $3 x + 6 = - 12$

Step 2: Solve both cases.

For $3 x + 6 = 12$ :
$3 x = 12 - 6 = 6$
$x = \frac{6}{3} = 2$

For $3 x + 6 = - 12$ :
$3 x = - 12 - 6 = - 18$
$x = \frac{- 18}{3} = - 6$

Thus, the solutions are $x = 2$ and $x = - 6$ .

Example 42: Solve $| 4 x - 8 | = 9$

Solution:

Step 1: Split into two cases.

$4 x - 8 = 9$ or $4 x - 8 = - 9$

Step 2: Solve both cases.

For $4 x - 8 = 9$ :
$4 x = 9 + 8 = 17$
$x = \frac{17}{4}$

For $4 x - 8 = - 9$ :
$4 x = - 9 + 8 = - 1$
$x = \frac{- 1}{4}$

Thus, the solutions are $x = \frac{17}{4}$ and $x = \frac{- 1}{4}$ .

Example 43: Solve $| 5 x + 9 | = 14$

Solution:

Step 1: Split into two cases.

$5 x + 9 = 14$ or $5 x + 9 = - 14$

Step 2: Solve both cases.

For $5 x + 9 = 14$ :
$5 x = 14 - 9 = 5$
$x = \frac{5}{5} = 1$

For $5 x + 9 = - 14$ :
$5 x = - 14 - 9 = - 23$
$x = \frac{- 23}{5}$

Thus, the solutions are $x = 1$ and $x = \frac{- 23}{5}$ .

Example 44: Solve $| x - 9 | = 12$

Solution:

Step 1: Split into two cases.

$x - 9 = 12$ or $x - 9 = - 12$

Step 2: Solve both cases.

For $x - 9 = 12$ :
$x = 12 + 9 = 21$

For $x - 9 = - 12$ :
$x = - 12 + 9 = - 3$

Thus, the solutions are $x = 21$ and $x = - 3$ .

Example 45: Solve $| 2 x + 3 | = 10$

Solution:

Step 1: Split into two cases.

$2 x + 3 = 10$ or $2 x + 3 = - 10$

Step 2: Solve both cases.

For $2 x + 3 = 10$ :
$2 x = 10 - 3 = 7$
$x = \frac{7}{2}$

For $2 x + 3 = - 10$ :
$2 x = - 10 - 3 = - 13$
$x = \frac{- 13}{2}$

Thus, the solutions are $x = \frac{7}{2}$ and $x = \frac{- 13}{2}$ .

Example 46: Solve $| 4 x - 5 | = 15$

Solution:

Step 1: Split into two cases.

$4 x - 5 = 15$ or $4 x - 5 = - 15$

Step 2: Solve both cases.

For $4 x - 5 = 15$ :
$4 x = 15 + 5 = 20$
$x = \frac{20}{4} = 5$

For $4 x - 5 = - 15$ :
$4 x = - 15 + 5 = - 10$
$x = \frac{- 10}{4} = - 2.5$

Thus, the solutions are $x = 5$ and $x = - 2.5$ .

Example 47: Solve $| x + 10 | = 13$

Solution:

Step 1: Split into two cases.

$x + 10 = 13$ or $x + 10 = - 13$

Step 2: Solve both cases.

For $x + 10 = 13$ :
$x = 13 - 10 = 3$

For $x + 10 = - 13$ :
$x = - 13 - 10 = - 23$

Thus, the solutions are $x = 3$ and $x = - 23$ .

Example 48: Solve $| 3 x - 7 | = 11$

Solution:

Step 1: Split into two cases.

$3 x - 7 = 11$ or $3 x - 7 = - 11$

Step 2: Solve both cases.

For $3 x - 7 = 11$ :
$3 x = 11 + 7 = 18$
$x = \frac{18}{3} = 6$

For $3 x - 7 = - 11$ :
$3 x = - 11 + 7 = - 4$
$x = \frac{- 4}{3}$

Thus, the solutions are $x = 6$ and $x = \frac{- 4}{3}$ .

Example 49: Solve $| 6 x + 1 | = 9$

Solution:

Step 1: Split into two cases.

$6 x + 1 = 9$ or $6 x + 1 = - 9$

Step 2: Solve both cases.

For $6 x + 1 = 9$ :
$6 x = 9 - 1 = 8$
$x = \frac{8}{6} = \frac{4}{3}$

For $6 x + 1 = - 9$ :
$6 x = - 9 - 1 = - 10$
$x = \frac{- 10}{6} = \frac{- 5}{3}$

Thus, the solutions are $x = \frac{4}{3}$ and $x = \frac{- 5}{3}$ .

Example 50: Solve $| 5 x - 8 | = 3$

Solution:

Step 1: Split into two cases.

$5 x - 8 = 3$ or $5 x - 8 = - 3$

Step 2: Solve both cases.

For $5 x - 8 = 3$ :
$5 x = 3 + 8 = 11$
$x = \frac{11}{5}$

For $5 x - 8 = - 3$ :
$5 x = - 3 + 8 = 5$
$x = \frac{5}{5} = 1$

Thus, the solutions are $x = \frac{11}{5}$ and $x = 1$ .

Example 51: Solve $| 7 x + 3 | = 10$

Solution:

Step 1: Split into two cases.

$7 x + 3 = 10$ or $7 x + 3 = - 10$

Step 2: Solve both cases.

For $7 x + 3 = 10$ :
$7 x = 10 - 3 = 7$
$x = \frac{7}{7} = 1$

For $7 x + 3 = - 10$ :
$7 x = - 10 - 3 = - 13$
$x = \frac{- 13}{7}$

Thus, the solutions are $x = 1$ and $x = \frac{- 13}{7}$ .

Example 52: Solve $| 2 x - 6 | = 4$

Solution:

Step 1: Split into two cases.

$2 x - 6 = 4$ or $2 x - 6 = - 4$

Step 2: Solve both cases.

For $2 x - 6 = 4$ :
$2 x = 4 + 6 = 10$
$x = \frac{10}{2} = 5$

For $2 x - 6 = - 4$ :
$2 x = - 4 + 6 = 2$
$x = \frac{2}{2} = 1$

Thus, the solutions are $x = 5$ and $x = 1$ .

Example 53: Solve $| 3 x + 7 | = 6$

Solution:

Step 1: Split into two cases.

$3 x + 7 = 6$ or $3 x + 7 = - 6$

Step 2: Solve both cases.

For $3 x + 7 = 6$ :
$3 x = 6 - 7 = - 1$
$x = \frac{- 1}{3}$

For $3 x + 7 = - 6$ :
$3 x = - 6 - 7 = - 13$
$x = \frac{- 13}{3}$

Thus, the solutions are $x = \frac{- 1}{3}$ and $x = \frac{- 13}{3}$ .

Example 54: Solve $| x - 2 | = 11$

Solution:

Step 1: Split into two cases.

$x - 2 = 11$ or $x - 2 = - 11$

Step 2: Solve both cases.

For $x - 2 = 11$ :
$x = 11 + 2 = 13$

For $x - 2 = - 11$ :
$x = - 11 + 2 = - 9$

Thus, the solutions are $x = 13$ and $x = - 9$ .

Example 55: Solve $| 4 x + 5 | = 13$

Solution:

Step 1: Split into two cases.

$4 x + 5 = 13$ or $4 x + 5 = - 13$

Step 2: Solve both cases.

For $4 x + 5 = 13$ :
$4 x = 13 - 5 = 8$
$x = \frac{8}{4} = 2$

For $4 x + 5 = - 13$ :
$4 x = - 13 - 5 = - 18$
$x = \frac{- 18}{4} = - 4.5$

Thus, the solutions are $x = 2$ and $x = - 4.5$ .

Example 56: Solve $| 5 x - 4 | = 7$

Solution:

Step 1: Split into two cases.

$5 x - 4 = 7$ or $5 x - 4 = - 7$

Step 2: Solve both cases.

For $5 x - 4 = 7$ :
$5 x = 7 + 4 = 11$
$x = \frac{11}{5}$

For $5 x - 4 = - 7$ :
$5 x = - 7 + 4 = - 3$
$x = \frac{- 3}{5}$

Thus, the solutions are $x = \frac{11}{5}$ and $x = \frac{- 3}{5}$ .

Example 57: Solve $| 2 x + 9 | = 1$

Solution:

Step 1: Split into two cases.

$2 x + 9 = 1$ or $2 x + 9 = - 1$

Step 2: Solve both cases.

For $2 x + 9 = 1$ :
$2 x = 1 - 9 = - 8$
$x = \frac{- 8}{2} = - 4$

For $2 x + 9 = - 1$ :
$2 x = - 1 - 9 = - 10$
$x = \frac{- 10}{2} = - 5$

Thus, the solutions are $x = - 4$ and $x = - 5$ .

Example 58: Solve $| 6 x - 3 | = 15$

Solution:

Step 1: Split into two cases.

$6 x - 3 = 15$ or $6 x - 3 = - 15$

Step 2: Solve both cases.

For $6 x - 3 = 15$ :
$6 x = 15 + 3 = 18$
$x = \frac{18}{6} = 3$

For $6 x - 3 = - 15$ :
$6 x = - 15 + 3 = - 12$
$x = \frac{- 12}{6} = - 2$

Thus, the solutions are $x = 3$ and $x = - 2$ .

Example 59: Solve $| x + 5 | = 8$

Solution:

Step 1: Split into two cases.

$x + 5 = 8$ or $x + 5 = - 8$

Step 2: Solve both cases.

For $x + 5 = 8$ :
$x = 8 - 5 = 3$

For $x + 5 = - 8$ :
$x = - 8 - 5 = - 13$

Thus, the solutions are $x = 3$ and $x = - 13$ .

Example 60: Solve $| 7 x + 4 | = 18$

Solution:

Step 1: Split into two cases.

$7 x + 4 = 18$ or $7 x + 4 = - 18$

Step 2: Solve both cases.

For $7 x + 4 = 18$ :
$7 x = 18 - 4 = 14$
$x = \frac{14}{7} = 2$

For $7 x + 4 = - 18$ :
$7 x = - 18 - 4 = - 22$
$x = \frac{- 22}{7}$

Thus, the solutions are $x = 2$ and $x = \frac{- 22}{7}$ .

Example 61: Solve $| 2 x - 7 | = 3$

Solution:

Step 1: Split into two cases.

$2 x - 7 = 3$ or $2 x - 7 = - 3$

Step 2: Solve both cases.

For $2 x - 7 = 3$ :
$2 x = 3 + 7 = 10$
$x = \frac{10}{2} = 5$

For $2 x - 7 = - 3$ :
$2 x = - 3 + 7 = 4$
$x = \frac{4}{2} = 2$

Thus, the solutions are $x = 5$ and $x = 2$ .

Example 62: Solve $| 5 x + 6 | = 12$

Solution:

Step 1: Split into two cases.

$5 x + 6 = 12$ or $5 x + 6 = - 12$

Step 2: Solve both cases.

For $5 x + 6 = 12$ :
$5 x = 12 - 6 = 6$
$x = \frac{6}{5}$

For $5 x + 6 = - 12$ :
$5 x = - 12 - 6 = - 18$
$x = \frac{- 18}{5}$

Thus, the solutions are $x = \frac{6}{5}$ and $x = \frac{- 18}{5}$ .

Example 63: Solve $| x + 7 | = 2$

Solution:

Step 1: Split into two cases.

$x + 7 = 2$ or $x + 7 = - 2$

Step 2: Solve both cases.

For $x + 7 = 2$ :
$x = 2 - 7 = - 5$

For $x + 7 = - 2$ :
$x = - 2 - 7 = - 9$

Thus, the solutions are $x = - 5$ and $x = - 9$ .

Example 64: Solve $| 4 x - 9 | = 8$

Solution:

Step 1: Split into two cases.

$4 x - 9 = 8$ or $4 x - 9 = - 8$

Step 2: Solve both cases.

For $4 x - 9 = 8$ :
$4 x = 8 + 9 = 17$
$x = \frac{17}{4}$

For $4 x - 9 = - 8$ :
$4 x = - 8 + 9 = 1$
$x = \frac{1}{4}$

Thus, the solutions are $x = \frac{17}{4}$ and $x = \frac{1}{4}$ .

Example 65: Solve $| 3 x + 1 | = 7$

Solution:

Step 1: Split into two cases.

$3 x + 1 = 7$ or $3 x + 1 = - 7$

Step 2: Solve both cases.

For $3 x + 1 = 7$ :
$3 x = 7 - 1 = 6$
$x = \frac{6}{3} = 2$

For $3 x + 1 = - 7$ :
$3 x = - 7 - 1 = - 8$
$x = \frac{- 8}{3}$

Thus, the solutions are $x = 2$ and $x = \frac{- 8}{3}$ .

Example 66: Solve $| 2 x - 3 | = 1$

Solution:

Step 1: Split into two cases.

$2 x - 3 = 1$ or $2 x - 3 = - 1$

Step 2: Solve both cases.

For $2 x - 3 = 1$ :
$2 x = 1 + 3 = 4$
$x = \frac{4}{2} = 2$

For $2 x - 3 = - 1$ :
$2 x = - 1 + 3 = 2$
$x = \frac{2}{2} = 1$

Thus, the solutions are $x = 2$ and $x = 1$ .

Example 67: Solve $| 6 x + 5 | = 13$

Solution:

Step 1: Split into two cases.

$6 x + 5 = 13$ or $6 x + 5 = - 13$

Step 2: Solve both cases.

For $6 x + 5 = 13$ :
$6 x = 13 - 5 = 8$
$x = \frac{8}{6} = \frac{4}{3}$

For $6 x + 5 = - 13$ :
$6 x = - 13 - 5 = - 18$
$x = \frac{- 18}{6} = - 3$

Thus, the solutions are $x = \frac{4}{3}$ and $x = - 3$ .

Example 68: Solve $| x - 4 | = 6$

Solution:

Step 1: Split into two cases.

$x - 4 = 6$ or $x - 4 = - 6$

Step 2: Solve both cases.

For $x - 4 = 6$ :
$x = 6 + 4 = 10$

For $x - 4 = - 6$ :
$x = - 6 + 4 = - 2$

Thus, the solutions are $x = 10$ and $x = - 2$ .

Example 69: Solve $| 5 x - 7 | = 2$

Solution:

Step 1: Split into two cases.

$5 x - 7 = 2$ or $5 x - 7 = - 2$

Step 2: Solve both cases.

For $5 x - 7 = 2$ :
$5 x = 2 + 7 = 9$
$x = \frac{9}{5}$

For $5 x - 7 = - 2$ :
$5 x = - 2 + 7 = 5$
$x = \frac{5}{5} = 1$

Thus, the solutions are $x = \frac{9}{5}$ and $x = 1$ .

Example 70: Solve $| x + 6 | = 8$

Solution:

Step 1: Split into two cases.

$x + 6 = 8$ or $x + 6 = - 8$

Step 2: Solve both cases.

For $x + 6 = 8$ :
$x = 8 - 6 = 2$

For $x + 6 = - 8$ :
$x = - 8 - 6 = - 14$

Thus, the solutions are $x = 2$ and $x = - 14$ .

Example 71: Solve $| 3 x + 4 | = 5$

Solution:

Step 1: Split into two cases.

$3 x + 4 = 5$ or $3 x + 4 = - 5$

Step 2: Solve both cases.

For $3 x + 4 = 5$ :
$3 x = 5 - 4 = 1$
$x = \frac{1}{3}$

For $3 x + 4 = - 5$ :
$3 x = - 5 - 4 = - 9$
$x = \frac{- 9}{3} = - 3$

Thus, the solutions are $x = \frac{1}{3}$ and $x = - 3$ .

Example 72: Solve $| 7 x - 2 | = 3$

Solution:

Step 1: Split into two cases.

$7 x - 2 = 3$ or $7 x - 2 = - 3$

Step 2: Solve both cases.

For $7 x - 2 = 3$ :
$7 x = 3 + 2 = 5$
$x = \frac{5}{7}$

For $7 x - 2 = - 3$ :
$7 x = - 3 + 2 = - 1$
$x = \frac{- 1}{7}$

Thus, the solutions are $x = \frac{5}{7}$ and $x = \frac{- 1}{7}$ .

Example 73: Solve $| x + 9 | = 10$

Solution:

Step 1: Split into two cases.

$x + 9 = 10$ or $x + 9 = - 10$

Step 2: Solve both cases.

For $x + 9 = 10$ :
$x = 10 - 9 = 1$

For $x + 9 = - 10$ :
$x = - 10 - 9 = - 19$

Thus, the solutions are $x = 1$ and $x = - 19$ .

Example 74: Solve $| 4 x - 6 | = 2$

Solution:

Step 1: Split into two cases.

$4 x - 6 = 2$ or $4 x - 6 = - 2$

Step 2: Solve both cases.

For $4 x - 6 = 2$ :
$4 x = 2 + 6 = 8$
$x = \frac{8}{4} = 2$

For $4 x - 6 = - 2$ :
$4 x = - 2 + 6 = 4$
$x = \frac{4}{4} = 1$

Thus, the solutions are $x = 2$ and $x = 1$ .

Example 75: Solve $| x - 5 | = 7$

Solution:

Step 1: Split into two cases.

$x - 5 = 7$ or $x - 5 = - 7$

Step 2: Solve both cases.

For $x - 5 = 7$ :
$x = 7 + 5 = 12$

For $x - 5 = - 7$ :
$x = - 7 + 5 = - 2$

Thus, the solutions are $x = 12$ and $x = - 2$ .

Example 76: Solve $| 2 x + 5 | = 3$

Solution:

Step 1: Split into two cases.

$2 x + 5 = 3$ or $2 x + 5 = - 3$

Step 2: Solve both cases.

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

For $2 x + 5 = - 3$ :
$2 x = - 3 - 5 = - 8$
$x = \frac{- 8}{2} = - 4$

Thus, the solutions are $x = - 1$ and $x = - 4$ .

Example 77: Solve $| 3 x - 4 | = 6$

Solution:

Step 1: Split into two cases.

$3 x - 4 = 6$ or $3 x - 4 = - 6$

Step 2: Solve both cases.

For $3 x - 4 = 6$ :
$3 x = 6 + 4 = 10$
$x = \frac{10}{3}$

For $3 x - 4 = - 6$ :
$3 x = - 6 + 4 = - 2$
$x = \frac{- 2}{3}$

Thus, the solutions are $x = \frac{10}{3}$ and $x = \frac{- 2}{3}$ .

Example 78: Solve $| x + 8 | = 6$

Solution:

Step 1: Split into two cases.

$x + 8 = 6$ or $x + 8 = - 6$

Step 2: Solve both cases.

For $x + 8 = 6$ :
$x = 6 - 8 = - 2$

For $x + 8 = - 6$ :
$x = - 6 - 8 = - 14$

Thus, the solutions are $x = - 2$ and $x = - 14$ .

Example 79: Solve $| 5 x - 3 | = 7$

Solution:

Step 1: Split into two cases.

$5 x - 3 = 7$ or $5 x - 3 = - 7$

Step 2: Solve both cases.

For $5 x - 3 = 7$ :
$5 x = 7 + 3 = 10$
$x = \frac{10}{5} = 2$

For $5 x - 3 = - 7$ :
$5 x = - 7 + 3 = - 4$
$x = \frac{- 4}{5}$

Thus, the solutions are $x = 2$ and $x = \frac{- 4}{5}$ .

Example 80: Solve $| 7 x + 2 | = 5$

Solution:

Step 1: Split into two cases.

$7 x + 2 = 5$ or $7 x + 2 = - 5$

Step 2: Solve both cases.

For $7 x + 2 = 5$ :
$7 x = 5 - 2 = 3$
$x = \frac{3}{7}$

For $7 x + 2 = - 5$ :
$7 x = - 5 - 2 = - 7$
$x = \frac{- 7}{7} = - 1$

Thus, the solutions are $x = \frac{3}{7}$ and $x = - 1$ .

Example 81: Solve $| x - 6 | = 4$

Solution:

Step 1: Split into two cases.

$x - 6 = 4$ or $x - 6 = - 4$

Step 2: Solve both cases.

For $x - 6 = 4$ :
$x = 4 + 6 = 10$

For $x - 6 = - 4$ :
$x = - 4 + 6 = 2$

Thus, the solutions are $x = 10$ and $x = 2$ .

Example 82: Solve $| 4 x + 9 | = 15$

Solution:

Step 1: Split into two cases.

$4 x + 9 = 15$ or $4 x + 9 = - 15$

Step 2: Solve both cases.

For $4 x + 9 = 15$ :
$4 x = 15 - 9 = 6$
$x = \frac{6}{4} = \frac{3}{2}$

For $4 x + 9 = - 15$ :
$4 x = - 15 - 9 = - 24$
$x = \frac{- 24}{4} = - 6$

Thus, the solutions are $x = \frac{3}{2}$ and $x = - 6$ .

Example 83: Solve $| 3 x - 5 | = 2$

Solution:

Step 1: Split into two cases.

$3 x - 5 = 2$ or $3 x - 5 = - 2$

Step 2: Solve both cases.

For $3 x - 5 = 2$ :
$3 x = 2 + 5 = 7$
$x = \frac{7}{3}$

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

Thus, the solutions are $x = \frac{7}{3}$ and $x = 1$ .

Example 84: Solve $| x + 4 | = 9$

Solution:

Step 1: Split into two cases.

$x + 4 = 9$ or $x + 4 = - 9$

Step 2: Solve both cases.

For $x + 4 = 9$ :
$x = 9 - 4 = 5$

For $x + 4 = - 9$ :
$x = - 9 - 4 = - 13$

Thus, the solutions are $x = 5$ and $x = - 13$ .

Example 85: Solve $| 5 x + 7 | = 12$

Solution:

Step 1: Split into two cases.

$5 x + 7 = 12$ or $5 x + 7 = - 12$

Step 2: Solve both cases.

For $5 x + 7 = 12$ :
$5 x = 12 - 7 = 5$
$x = \frac{5}{5} = 1$

For $5 x + 7 = - 12$ :
$5 x = - 12 - 7 = - 19$
$x = \frac{- 19}{5}$

Thus, the solutions are $x = 1$ and $x = \frac{- 19}{5}$ .

Example 86: Solve $| 2 x - 8 | = 6$

Solution:

Step 1: Split into two cases.

$2 x - 8 = 6$ or $2 x - 8 = - 6$

Step 2: Solve both cases.

For $2 x - 8 = 6$ :
$2 x = 6 + 8 = 14$
$x = \frac{14}{2} = 7$

For $2 x - 8 = - 6$ :
$2 x = - 6 + 8 = 2$
$x = \frac{2}{2} = 1$

Thus, the solutions are $x = 7$ and $x = 1$ .

Example 87: Solve $| 4 x + 1 | = 11$

Solution:

Step 1: Split into two cases.

$4 x + 1 = 11$ or $4 x + 1 = - 11$

Step 2: Solve both cases.

For $4 x + 1 = 11$ :
$4 x = 11 - 1 = 10$
$x = \frac{10}{4} = 2.5$

For $4 x + 1 = - 11$ :
$4 x = - 11 - 1 = - 12$
$x = \frac{- 12}{4} = - 3$

Thus, the solutions are $x = 2.5$ and $x = - 3$ .

Example 88: Solve $| 3 x - 2 | = 4$

Solution:

Step 1: Split into two cases.

$3 x - 2 = 4$ or $3 x - 2 = - 4$

Step 2: Solve both cases.

For $3 x - 2 = 4$ :
$3 x = 4 + 2 = 6$
$x = \frac{6}{3} = 2$

For $3 x - 2 = - 4$ :
$3 x = - 4 + 2 = - 2$
$x = \frac{- 2}{3}$

Thus, the solutions are $x = 2$ and $x = \frac{- 2}{3}$ .

Example 89: Solve $| x + 10 | = 14$

Solution:

Step 1: Split into two cases.

$x + 10 = 14$ or $x + 10 = - 14$

Step 2: Solve both cases.

For $x + 10 = 14$ :
$x = 14 - 10 = 4$

For $x + 10 = - 14$ :
$x = - 14 - 10 = - 24$

Thus, the solutions are $x = 4$ and $x = - 24$ .

Example 90: Solve $| 2 x + 3 | = 9$

Solution:

Step 1: Split into two cases.

$2 x + 3 = 9$ or $2 x + 3 = - 9$

Step 2: Solve both cases.

For $2 x + 3 = 9$ :
$2 x = 9 - 3 = 6$
$x = \frac{6}{2} = 3$

For $2 x + 3 = - 9$ :
$2 x = - 9 - 3 = - 12$
$x = \frac{- 12}{2} = - 6$

Thus, the solutions are $x = 3$ and $x = - 6$ .

Example 91: Solve $| 6 x - 4 | = 14$

Solution:

Step 1: Split into two cases.

$6 x - 4 = 14$ or $6 x - 4 = - 14$

Step 2: Solve both cases.

For $6 x - 4 = 14$ :
$6 x = 14 + 4 = 18$
$x = \frac{18}{6} = 3$

For $6 x - 4 = - 14$ :
$6 x = - 14 + 4 = - 10$
$x = \frac{- 10}{6} = \frac{- 5}{3}$

Thus, the solutions are $x = 3$ and $x = \frac{- 5}{3}$ .

Example 92: Solve $| 7 x + 9 | = 16$

Solution:

Step 1: Split into two cases.

$7 x + 9 = 16$ or $7 x + 9 = - 16$

Step 2: Solve both cases.

For $7 x + 9 = 16$ :
$7 x = 16 - 9 = 7$
$x = \frac{7}{7} = 1$

For $7 x + 9 = - 16$ :
$7 x = - 16 - 9 = - 25$
$x = \frac{- 25}{7}$

Thus, the solutions are $x = 1$ and $x = \frac{- 25}{7}$ .

Example 93: Solve $| x + 6 | = 11$

Solution:

Step 1: Split into two cases.

$x + 6 = 11$ or $x + 6 = - 11$

Step 2: Solve both cases.

For $x + 6 = 11$ :
$x = 11 - 6 = 5$

For $x + 6 = - 11$ :
$x = - 11 - 6 = - 17$

Thus, the solutions are $x = 5$ and $x = - 17$ .

Example 94: Solve $| 4 x - 10 | = 12$

Solution:

Step 1: Split into two cases.

$4 x - 10 = 12$ or $4 x - 10 = - 12$

Step 2: Solve both cases.

For $4 x - 10 = 12$ :
$4 x = 12 + 10 = 22$
$x = \frac{22}{4} = \frac{11}{2}$

For $4 x - 10 = - 12$ :
$4 x = - 12 + 10 = - 2$
$x = \frac{- 2}{4} = \frac{- 1}{2}$

Thus, the solutions are $x = \frac{11}{2}$ and $x = \frac{- 1}{2}$ .

Example 95: Solve $| 3 x + 8 | = 7$

Solution:

Step 1: Split into two cases.

$3 x + 8 = 7$ or $3 x + 8 = - 7$

Step 2: Solve both cases.

For $3 x + 8 = 7$ :
$3 x = 7 - 8 = - 1$
$x = \frac{- 1}{3}$

For $3 x + 8 = - 7$ :
$3 x = - 7 - 8 = - 15$
$x = \frac{- 15}{3} = - 5$

Thus, the solutions are $x = \frac{- 1}{3}$ and $x = - 5$ .

Example 96: Solve $| 2 x - 4 | = 7$

Solution:

Step 1: Split into two cases.

$2 x - 4 = 7$ or $2 x - 4 = - 7$

Step 2: Solve both cases.

For $2 x - 4 = 7$ :
$2 x = 7 + 4 = 11$
$x = \frac{11}{2}$

For $2 x - 4 = - 7$ :
$2 x = - 7 + 4 = - 3$
$x = \frac{- 3}{2}$

Thus, the solutions are $x = \frac{11}{2}$ and $x = \frac{- 3}{2}$ .

Example 97: Solve $| 5 x + 2 | = 9$

Solution:

Step 1: Split into two cases.

$5 x + 2 = 9$ or $5 x + 2 = - 9$

Step 2: Solve both cases.

For $5 x + 2 = 9$ :
$5 x = 9 - 2 = 7$
$x = \frac{7}{5}$

For $5 x + 2 = - 9$ :
$5 x = - 9 - 2 = - 11$
$x = \frac{- 11}{5}$

Thus, the solutions are $x = \frac{7}{5}$ and $x = \frac{- 11}{5}$ .

Example 98: Solve $| x + 12 | = 8$

Solution:

Step 1: Split into two cases.

$x + 12 = 8$ or $x + 12 = - 8$

Step 2: Solve both cases.

For $x + 12 = 8$ :
$x = 8 - 12 = - 4$

For $x + 12 = - 8$ :
$x = - 8 - 12 = - 20$

Thus, the solutions are $x = - 4$ and $x = - 20$ .

Example 99: Solve $| 6 x - 1 | = 4$

Solution:

Step 1: Split into two cases.

$6 x - 1 = 4$ or $6 x - 1 = - 4$

Step 2: Solve both cases.

For $6 x - 1 = 4$ :
$6 x = 4 + 1 = 5$
$x = \frac{5}{6}$

For $6 x - 1 = - 4$ :
$6 x = - 4 + 1 = - 3$
$x = \frac{- 3}{6} = \frac{- 1}{2}$

Thus, the solutions are $x = \frac{5}{6}$ and $x = \frac{- 1}{2}$ .

Example 100: Solve $| 4 x + 7 | = 11$

Solution:

Step 1: Split into two cases.

$4 x + 7 = 11$ or $4 x + 7 = - 11$

Step 2: Solve both cases.

For $4 x + 7 = 11$ :
$4 x = 11 - 7 = 4$
$x = \frac{4}{4} = 1$

For $4 x + 7 = - 11$ :
$4 x = - 11 - 7 = - 18$
$x = \frac{- 18}{4} = - 4.5$

Thus, the solutions are $x = 1$ and $x = - 4.5$ .

Introduction to Absolute Value Functions

Example 1: Solve |x|=5

Example 2: Solve |x+3|=7

Example 3: Solve |2x–1|=9

Example 4: Solve |3x+2|=0

Example 5: Solve |x–4|=3

Example 6: Solve |2x+5|=10

Example 7: Solve |x+6|=4

Example 8: Solve |4x–7|=12

Example 9: Solve |x–5|=8

Example 10: Solve |3x+4|=7

Example 11: Solve |x+9|=6

Example 12: Solve |2x–3|=5

Example 13: Solve |x–7|=9

Example 14: Solve |2x+1|=11

Example 15: Solve |x–2|=6

Example 16: Solve |5x–3|=13

Example 17: Solve |x+5|=12

Example 18: Solve |4x–9|=7

Example 19: Solve |2x–4|=8

Example 20: Solve |3x+1|=10

Example 21: Solve |5x+3|=15

Example 22: Solve |x–8|=4

Example 23: Solve |x+4|=5

Example 24: Solve |2x+1|=9

Example 25: Solve |3x–2|=6

Example 26: Solve |x–1|=9

Example 27: Solve |4x–3|=2

Example 28: Solve |x+6|=3

Example 29: Solve |2x–1|=7

Example 30: Solve |5x+4|=14

Example 31: Solve |x+3|=5

Example 32: Solve |4x–6|=10

Example 33: Solve |3x+2|=9

Example 34: Solve |x–10|=7

Example 35: Solve |2x+7|=5

Example 36: Solve |5x–2|=8

Example 37: Solve |x+4|=9

Example 38: Solve |6x–5|=11

Example 39: Solve |x–3|=10

Example 40: Solve |7x+2|=16

Example 41: Solve |3x+6|=12

Example 42: Solve |4x–8|=9

Example 43: Solve |5x+9|=14

Example 44: Solve |x–9|=12

Example 45: Solve |2x+3|=10

Example 46: Solve |4x–5|=15

Example 47: Solve |x+10|=13

Example 48: Solve |3x–7|=11

Example 49: Solve |6x+1|=9

Example 50: Solve |5x–8|=3

Example 51: Solve |7x+3|=10

Example 52: Solve |2x–6|=4

Example 53: Solve |3x+7|=6

Example 54: Solve |x–2|=11

Example 55: Solve |4x+5|=13

Example 56: Solve |5x–4|=7

Example 57: Solve |2x+9|=1

Example 58: Solve |6x–3|=15

Example 59: Solve |x+5|=8

Example 60: Solve |7x+4|=18

Example 61: Solve |2x–7|=3

Example 62: Solve |5x+6|=12

Example 63: Solve |x+7|=2

Example 64: Solve |4x–9|=8

Example 65: Solve |3x+1|=7

Example 66: Solve |2x–3|=1

Example 67: Solve |6x+5|=13

Example 68: Solve |x–4|=6

Example 69: Solve |5x–7|=2

Example 70: Solve |x+6|=8

Example 71: Solve |3x+4|=5

Example 72: Solve |7x–2|=3

Example 73: Solve |x+9|=10

Example 74: Solve |4x–6|=2

Example 75: Solve |x–5|=7

Example 76: Solve |2x+5|=3

Example 77: Solve |3x–4|=6

Example 78: Solve |x+8|=6

Example 79: Solve |5x–3|=7

Example 1: Solve $| x | = 5$

Example 2: Solve $| x + 3 | = 7$

Example 3: Solve $| 2 x - 1 | = 9$

Example 4: Solve $| 3 x + 2 | = 0$

Example 5: Solve $| x - 4 | = 3$

Example 6: Solve $| 2 x + 5 | = 10$

Example 7: Solve $| x + 6 | = 4$

Example 8: Solve $| 4 x - 7 | = 12$

Example 9: Solve $| x - 5 | = 8$

Example 10: Solve $| 3 x + 4 | = 7$

Example 11: Solve $| x + 9 | = 6$

Example 12: Solve $| 2 x - 3 | = 5$

Example 13: Solve $| x - 7 | = 9$

Example 14: Solve $| 2 x + 1 | = 11$

Example 15: Solve $| x - 2 | = 6$

Example 16: Solve $| 5 x - 3 | = 13$

Example 17: Solve $| x + 5 | = 12$

Example 18: Solve $| 4 x - 9 | = 7$

Example 19: Solve $| 2 x - 4 | = 8$

Example 20: Solve $| 3 x + 1 | = 10$

Example 21: Solve $| 5 x + 3 | = 15$

Example 22: Solve $| x - 8 | = 4$

Example 23: Solve $| x + 4 | = 5$

Example 24: Solve $| 2 x + 1 | = 9$

Example 25: Solve $| 3 x - 2 | = 6$

Example 26: Solve $| x - 1 | = 9$

Example 27: Solve $| 4 x - 3 | = 2$

Example 28: Solve $| x + 6 | = 3$

Example 29: Solve $| 2 x - 1 | = 7$

Example 30: Solve $| 5 x + 4 | = 14$

Example 31: Solve $| x + 3 | = 5$

Example 32: Solve $| 4 x - 6 | = 10$

Example 33: Solve $| 3 x + 2 | = 9$

Example 34: Solve $| x - 10 | = 7$

Example 35: Solve $| 2 x + 7 | = 5$

Example 36: Solve $| 5 x - 2 | = 8$

Example 37: Solve $| x + 4 | = 9$

Example 38: Solve $| 6 x - 5 | = 11$

Example 39: Solve $| x - 3 | = 10$

Example 40: Solve $| 7 x + 2 | = 16$

Example 41: Solve $| 3 x + 6 | = 12$

Example 42: Solve $| 4 x - 8 | = 9$

Example 43: Solve $| 5 x + 9 | = 14$

Example 44: Solve $| x - 9 | = 12$

Example 45: Solve $| 2 x + 3 | = 10$

Example 46: Solve $| 4 x - 5 | = 15$

Example 47: Solve $| x + 10 | = 13$

Example 48: Solve $| 3 x - 7 | = 11$

Example 49: Solve $| 6 x + 1 | = 9$

Example 50: Solve $| 5 x - 8 | = 3$

Example 51: Solve $| 7 x + 3 | = 10$

Example 52: Solve $| 2 x - 6 | = 4$

Example 53: Solve $| 3 x + 7 | = 6$

Example 54: Solve $| x - 2 | = 11$

Example 55: Solve $| 4 x + 5 | = 13$

Example 56: Solve $| 5 x - 4 | = 7$

Example 57: Solve $| 2 x + 9 | = 1$

Example 58: Solve $| 6 x - 3 | = 15$

Example 59: Solve $| x + 5 | = 8$

Example 60: Solve $| 7 x + 4 | = 18$

Example 61: Solve $| 2 x - 7 | = 3$

Example 62: Solve $| 5 x + 6 | = 12$

Example 63: Solve $| x + 7 | = 2$

Example 64: Solve $| 4 x - 9 | = 8$

Example 65: Solve $| 3 x + 1 | = 7$

Example 66: Solve $| 2 x - 3 | = 1$

Example 67: Solve $| 6 x + 5 | = 13$

Example 68: Solve $| x - 4 | = 6$

Example 69: Solve $| 5 x - 7 | = 2$

Example 70: Solve $| x + 6 | = 8$

Example 71: Solve $| 3 x + 4 | = 5$

Example 72: Solve $| 7 x - 2 | = 3$

Example 73: Solve $| x + 9 | = 10$

Example 74: Solve $| 4 x - 6 | = 2$

Example 75: Solve $| x - 5 | = 7$

Example 76: Solve $| 2 x + 5 | = 3$

Example 77: Solve $| 3 x - 4 | = 6$

Example 78: Solve $| x + 8 | = 6$

Example 79: Solve $| 5 x - 3 | = 7$

Example 80: Solve $| 7 x + 2 | = 5$