Absolute Value Inequalities

1.

$|4x-7|<9$

$$\begin {array}{cc}\eqalign{4x-7&< 9 \\ +7&\quad +7 \\4x&< 16\\\div 4 &\;  \div 4\\x&<4 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{4x-7 & > ^-9\\ +7&\quad +7\\4x& > -2\\\div 4 &\; \div 4\\x&> -\dfrac{1}{2}}\end{array}$$

2.

$|3x+3|\leq 9$

$$\begin {array}{cc}\eqalign{3x+3& \leq 9 \\ -3&\quad -3 \\3x&\leq 6\\\div 3 &\;  \div 3\\x&\leq 2 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{3x+3 & \geq ^-9\\ -3&\quad -3\\3x& \geq -12\\\div 3 &\; \div 3\\x&\geq -4}\end{array}$$

3.

$|2x-1|\geq 11$

$$\begin {array}{cc}\eqalign{2x-1& \geq 11 \\ +1&\quad +1 \\2x&\geq 12\\\div 2 &\;  \div 2\\x&\geq 6 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{2x-1 & \leq ^-11\\ +1&\quad +1\\2x& \leq -1\\\div 2 &\; \div 2\\x&\leq -5}\end{array}$$

4.

$\eqalign{|x+1|-1&>9\\+1 &\quad +1\\|x+1|&>10}$

$$\begin {array}{cc}\eqalign{x+1& > 10 \\ -1&\quad -1 \\x&> 9}\end {array}\qquad \qquad \begin{array}{cc}\eqalign{x+1 & < ^-10\\ -1&\quad -1\\x& < -11}\end{array}$$

5.

$\eqalign{|x+2|-3&>11\\ +3 &\quad +3\\|x+2|&>14}$

$$\begin {array}{cc}\eqalign{x+2& > 14 \\ -2&\quad -2 \\x&> 12}\end {array}\qquad \qquad \begin{array}{cc}\eqalign{x+2 & < ^-14\\ -2&\quad -2\\x& < -16}\end{array}$$

6.

$\eqalign{|5x-4|+1 &\leq 19\\ -1 &\quad -1 \\|5x-4|&\leq 18}$

$$\begin {array}{cc}\eqalign{5x-4& \leq 18 \\ +4&\quad +4 \\5x&\leq 22\\\div 5 &\;  \div 5\\x&\leq \dfrac{22}{5}}\end {array}\qquad \qquad \begin{array}{cc}\eqalign{5x-4 & \geq ^-18\\ +4&\quad +4\\5x& \geq -14\\\div 5 &\; \div 5\\x&\geq -\dfrac{14}{5}}\end{array}$$

7.

$\eqalign{2|x-2|&\leq 10\\ \div 2 &\quad \div 2\\|x-2|& \leq 5}$

$$\begin {array}{cc}\eqalign{x-2& \leq 5 \\ +2&\quad +2 \\x&\leq 7}\end {array}\qquad \qquad \begin{array}{cc}\eqalign{x-2 & \geq ^-5\\ +2&\quad +2\\x& \geq -3}\end{array}$$

8.

$\eqalign{3|2x-5|-1&<8\\+1 &\quad +1\\ 3|2x-5| &<9\\\div 3 &\quad \div 3\\|2x-5|& <3}$

$$\begin {array}{cc}\eqalign{2x-5& < 3 \\ +5&\quad +5 \\2x&< 8\\\div 2 &\;  \div 2\\x&< 4 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{2x-5 & > ^-3\\ +5&\quad +5\\2x& > 2\\\div 2 &\; \div 2\\x&> 1}\end{array}$$

9.

$\eqalign{2|2x-5|-1&<8\\+1 &\quad +1\\2|2x-5|&<9\\\div 2 &\quad \div 2\\|2x-5|&<\dfrac{9}{2}}$

$$\begin {array}{cc}\eqalign{2x-5& < \dfrac{9}{2} \\ +5&\quad +5 \\2x&< \dfrac{19}{2}\\\div 2 &\;  \div 2\\x&< \dfrac{19}{4} }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{2x-5 & > ^-\dfrac{9}{2}\\ +5&\quad +5\\2x& > \dfrac{1}{2}\\\div 2 &\; \div 2\\x&> \dfrac{1}{4}}\end{array}$$

10.

$\eqalign{4+2|2x-5|&<8\\-4 &\quad -4\\2|2x-5|&<4\\\div 2 &\quad \div 2\\|2x-5|&<2}$

$$\begin {array}{cc}\eqalign{2x-5& < 2 \\ +5&\quad +5 \\2x&< 7\\\div 2 &\;  \div 2\\x&< 3.5 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{2x-5 & > ^-2\\ +5&\quad +5\\2x& > 3\\\div 2 &\; \div 2\\x&>1.5 }\end{array}$$

11.

$\eqalign{3-1|2x-5|&<8\\-3 &\quad -3\\-1|2x-5|&<5\\\div -1 &\quad \div -1\\|2x-5|&>-5}$

$$\begin {array}{cc}\eqalign{2x-5& >^-5 \\ +5&\quad +5 \\2x&> 0\\\div 2 &\;  \div 2\\x&>0 }\end {array}\qquad \qquad \begin{array}{cc}\eqalign{2x-5 & < 5\\ +5&\quad +5\\2x& < 10\\\div 2 &\; \div 2\\x&<5 }\end{array}$$

12.

$\eqalign{5-4|2x-5|-3&<8\\2-4|2x-5|&<8\\-2 &\quad -2\\-4|2x-5|&<6\\\div -4 &\quad \div -4\\|2x-5|&>-\dfrac{3}{2}}$

$$\begin {array}{cc}\eqalign{2x-5& >-\dfrac{3}{2} \\ +5&\quad +5 \\2x&> -\dfrac{7}{2}\\\div 2 &\;  \div 2\\x&>-\dfrac{7}{4}}\end {array}\qquad \qquad \begin{array}{cc}\eqalign{2x-5 & < \dfrac{3}{2}\\ +5&\quad +5\\2x& < \dfrac{13}{2}\\\div 2 &\; \div 2\\x&<\dfrac{13}{4} }\end{array}$$