Solving absolute value inequalities can be very easy if you first isolate the absolute value portion to the left side of the equation, then choose which of these 9 scenarios you have.
\(\bullet \, |a| \lt b \) (b is positive) \(\,\,\,\Rightarrow\,\,\, -b \lt a \lt b\)
\(\bullet \, |a| \le b \) (b is positive) \(\,\,\,\Rightarrow\,\,\, -b \le a \le b\)
\(\bullet \, |a| \gt b \) (b is positive) \(\,\,\,\Rightarrow\,\,\, a \lt -b \) or \( a\gt b\)
\(\bullet \, |a| \ge b \) (b is positive) \(\,\,\,\Rightarrow\,\,\, a \le -b \) or \( a\ge b\)
\(\bullet \, |a| \gt c \) (c is negative) \(\,\,\,\Rightarrow\,\,\, \) All real Numbers
\(\bullet \, |a| \ge c \) (c is negative or 0) \(\,\,\,\Rightarrow\,\,\, \) All real Numbers
\(\bullet \, |a| \lt c \) (c is negative or 0) \(\,\,\,\Rightarrow\,\,\, \) No Solution
\(\bullet \, |a| \le c \) (c is negative) \(\,\,\,\Rightarrow\,\,\, \) No Solution
\(\bullet \, |a| \le 0 \) \(\,\,\,\Rightarrow\,\,\, a=0 \)
Practice Problems
Solve each absolute value inequality.
\(\textbf{1)}\) \( \left|5-2x\right|\le10 \)
The answer is \( -\displaystyle\frac{5}{2}\le x\le\displaystyle\frac{15}{2} \)
\(\,\,\,\,\,\,\left|5-2x\right|\le10 \)
\(\,\,\,\,\,\,-10 \le 5-2x \le10 \)
\(\,\,\,\,\,\,-15 \le -2x \le5 \)
\(\,\,\,\,\,\,\displaystyle \frac{-15}{-2} \ge x \ge \frac{5}{-2} \)
\(\,\,\,\,\,\,\displaystyle \frac{15}{2} \ge x \ge – \frac{5}{2} \)
\(\,\,\,\,\,\,-\displaystyle\frac{5}{2}\le x\le\displaystyle\frac{15}{2} \)
\(\,\,\,\,\,\,\)The answer is \( -\displaystyle\frac{5}{2}\le x\le\displaystyle\frac{15}{2} \)
\(\,\,\,\,\,\,\left|5-2x\right|\le10 \) \(\,\,\,\,\,\,-10 \le 5-2x \le10 \) \(\,\,\,\,\,\,-15 \le -2x \le5 \) \(\,\,\,\,\,\,\displaystyle \frac{-15}{-2} \ge x \ge \frac{5}{-2} \) \(\,\,\,\,\,\,\displaystyle \frac{15}{2} \ge x \ge – \frac{5}{2} \) \(\,\,\,\,\,\,-\displaystyle\frac{5}{2}\le x\le\displaystyle\frac{15}{2} \)
\(\,\,\,\,\,\,\)The answer is \( -\displaystyle\frac{5}{2}\le x\le\displaystyle\frac{15}{2} \)
\(\textbf{2)}\) \( 2\left|2x-3\right|\gt10 \)
The answer is \( x\lt-1 \) or \( x\gt4 \)
\(\textbf{3)}\) \( 3\left| x+1\right|+20\le5 \)
The answer is No Solution
\(\textbf{4)}\) \( 3\left| x\right|+4\ge-8 \)
The answer is All Real Numbers
\(\textbf{5)}\) \( 3\left| 2x-3\right|-5\le10 \)
The answer is \( -1\le x\le4 \)
\(\,\,\,\,\,\,3\left| 2x-3\right|-5\le10 \)
\(\,\,\,\,\,\,3\left| 2x-3\right|\le15 \)
\(\,\,\,\,\,\,\left| 2x-3\right|\le5 \)
\(\,\,\,\,\,\,-5\le 2x-3 \le5 \)
\(\,\,\,\,\,\,-2\le 2x \le8 \)
\(\,\,\,\,\,\,-1\le x \le4 \)
\(\,\,\,\,\,\,3\left| 2x-3\right|-5\le10 \)
\(\,\,\,\,\,\,3\left| 2x-3\right|\le15 \)
\(\,\,\,\,\,\,\left| 2x-3\right|\le5 \)
\(\,\,\,\,\,\,-5\le 2x-3 \le5 \)
\(\,\,\,\,\,\,-2\le 2x \le8 \)
\(\,\,\,\,\,\,-1\le x \le4 \)
\(\textbf{6)}\) \( \left| 3x-7\right|\le0 \)
The answer is \( x=\displaystyle\frac{7}{3} \)
\(\textbf{7)}\) \(\left| 5 – x \right|+1 \lt 5\)
The answer is \( 1\lt x \lt 9 \)
\(\,\,\,\,\,\,\left| 5 – x \right|+1 \lt 5\)
\(\,\,\,\,\,\,\left| 5 – x \right| \lt 4\)
\(\,\,\,\,\,\,-4 \lt 5 – x \lt 4\)
\(\,\,\,\,\,\,-9 \lt – x \lt -1\)
\(\,\,\,\,\,\,9 \gt x \gt 1\)
The answer is \( 1\lt x \lt 9 \)
\(\,\,\,\,\,\,\left| 5 – x \right|+1 \lt 5\)
\(\,\,\,\,\,\,\left| 5 – x \right| \lt 4\)
\(\,\,\,\,\,\,-4 \lt 5 – x \lt 4\)
\(\,\,\,\,\,\,-9 \lt – x \lt -1\)
\(\,\,\,\,\,\,9 \gt x \gt 1\)
The answer is \( 1\lt x \lt 9 \)
\(\textbf{8)}\) \(2\left| 1 – 2x \right| \le 10\)
The answer is \( -2\le x \le 3 \)
\(\textbf{9)}\) \(\left| 2x + 1 \right| \lt – 1\)
The answer is No Solution
\(\textbf{10)}\) \(\left| 1 – x \right| \gt 1\)
The answer is \(x \lt 0 \text{ or }x \gt 2\)
See Related Pages\(\)
\(\bullet\text{ Absolute Value Inequality Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Absolute Value Equations}\)
\(\,\,\,\,\,\,\,\,|2x+5|=11…\)
\(\bullet\text{ Andymath Homepage}\)
