Absolute value inequalities involve finding values that satisfy an inequality containing an absolute value expression. When the absolute value is less than a positive number, the solution is typically between two values, while greater-than inequalities usually have solutions outside an interval. Before solving, isolate the absolute value expression and determine whether the inequality has an “and” solution, an “or” solution, no solution, or all real numbers.
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 \(-\frac{5}{2}\le x\le\frac{15}{2}\)
\(\,\,\,\,\,\,\left|5-2x\right|\le10\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-10\le5-2x\le10\)
Subtract 5 from all three parts.
\(\,\,\,\,\,\,-15\le-2x\le5\)
Divide by -2 and reverse both inequality signs.
\(\,\,\,\,\,\,\frac{15}{2}\ge x\ge-\frac{5}{2}\)
Rewrite in increasing order.
\(\,\,\,\,\,\,-\frac{5}{2}\le x\le\frac{15}{2}\)
\(\,\,\,\,\,\,\left|5-2x\right|\le10\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-10\le5-2x\le10\)
Subtract 5 from all three parts.
\(\,\,\,\,\,\,-15\le-2x\le5\)
Divide by -2 and reverse both inequality signs.
\(\,\,\,\,\,\,\frac{15}{2}\ge x\ge-\frac{5}{2}\)
Rewrite in increasing order.
\(\,\,\,\,\,\,-\frac{5}{2}\le x\le\frac{15}{2}\)
\(\textbf{2)}\) \(2\left|2x-3\right|\gt10\) The answer is \(x\lt-1\) or \(x\gt4\)
\(\,\,\,\,\,\,2\left|2x-3\right|\gt10\)
Divide both sides by 2.
\(\,\,\,\,\,\,\left|2x-3\right|\gt5\)
Since the absolute value is greater than 5, split into two inequalities.
\(\,\,\,\,\,\,2x-3\lt-5\) or \(2x-3\gt5\)
Add 3 to both sides of each inequality.
\(\,\,\,\,\,\,2x\lt-2\) or \(2x\gt8\)
Divide by 2.
\(\,\,\,\,\,\,x\lt-1\) or \(x\gt4\)
\(\,\,\,\,\,\,2\left|2x-3\right|\gt10\)
Divide both sides by 2.
\(\,\,\,\,\,\,\left|2x-3\right|\gt5\)
Since the absolute value is greater than 5, split into two inequalities.
\(\,\,\,\,\,\,2x-3\lt-5\) or \(2x-3\gt5\)
Add 3 to both sides of each inequality.
\(\,\,\,\,\,\,2x\lt-2\) or \(2x\gt8\)
Divide by 2.
\(\,\,\,\,\,\,x\lt-1\) or \(x\gt4\)
\(\textbf{3)}\) \(3\left|x+1\right|+20\le5\) The answer is No Solution.
\(\,\,\,\,\,\,3\left|x+1\right|+20\le5\)
Subtract 20 from both sides.
\(\,\,\,\,\,\,3\left|x+1\right|\le-15\)
Divide both sides by 3.
\(\,\,\,\,\,\,\left|x+1\right|\le-5\)
An absolute value cannot be negative.
Therefore, there is no solution.
\(\,\,\,\,\,\,3\left|x+1\right|+20\le5\)
Subtract 20 from both sides.
\(\,\,\,\,\,\,3\left|x+1\right|\le-15\)
Divide both sides by 3.
\(\,\,\,\,\,\,\left|x+1\right|\le-5\)
An absolute value cannot be negative.
Therefore, there is no solution.
\(\textbf{4)}\) \(3\left|x\right|+4\ge-8\) The answer is All Real Numbers.
\(\,\,\,\,\,\,3\left|x\right|+4\ge-8\)
Subtract 4 from both sides.
\(\,\,\,\,\,\,3\left|x\right|\ge-12\)
Divide both sides by 3.
\(\,\,\,\,\,\,\left|x\right|\ge-4\)
An absolute value is always greater than or equal to zero.
Therefore, all real numbers satisfy the inequality.
\(\,\,\,\,\,\,3\left|x\right|+4\ge-8\)
Subtract 4 from both sides.
\(\,\,\,\,\,\,3\left|x\right|\ge-12\)
Divide both sides by 3.
\(\,\,\,\,\,\,\left|x\right|\ge-4\)
An absolute value is always greater than or equal to zero.
Therefore, all real numbers satisfy the inequality.
\(\textbf{5)}\) \(3\left|2x-3\right|-5\le10\) The answer is \(-1\le x\le4\)
\(\,\,\,\,\,\,3\left|2x-3\right|-5\le10\)
Add 5 to both sides.
\(\,\,\,\,\,\,3\left|2x-3\right|\le15\)
Divide both sides by 3.
\(\,\,\,\,\,\,\left|2x-3\right|\le5\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-5\le2x-3\le5\)
Add 3 to all three parts.
\(\,\,\,\,\,\,-2\le2x\le8\)
Divide all three parts by 2.
\(\,\,\,\,\,\,-1\le x\le4\)
\(\,\,\,\,\,\,3\left|2x-3\right|-5\le10\)
Add 5 to both sides.
\(\,\,\,\,\,\,3\left|2x-3\right|\le15\)
Divide both sides by 3.
\(\,\,\,\,\,\,\left|2x-3\right|\le5\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-5\le2x-3\le5\)
Add 3 to all three parts.
\(\,\,\,\,\,\,-2\le2x\le8\)
Divide all three parts by 2.
\(\,\,\,\,\,\,-1\le x\le4\)
\(\textbf{6)}\) \(\left|3x-7\right|\le0\) The answer is \(x=\frac{7}{3}\)
\(\,\,\,\,\,\,\left|3x-7\right|\le0\)
An absolute value is never negative, so it must equal zero.
\(\,\,\,\,\,\,3x-7=0\)
Add 7 to both sides.
\(\,\,\,\,\,\,3x=7\)
Divide both sides by 3.
\(\,\,\,\,\,\,x=\frac{7}{3}\)
\(\,\,\,\,\,\,\left|3x-7\right|\le0\)
An absolute value is never negative, so it must equal zero.
\(\,\,\,\,\,\,3x-7=0\)
Add 7 to both sides.
\(\,\,\,\,\,\,3x=7\)
Divide both sides by 3.
\(\,\,\,\,\,\,x=\frac{7}{3}\)
\(\textbf{7)}\) \(\left|5-x\right|+1\lt5\) The answer is \(1\lt x\lt9\)
\(\,\,\,\,\,\,\left|5-x\right|+1\lt5\)
Subtract 1 from both sides.
\(\,\,\,\,\,\,\left|5-x\right|\lt4\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-4\lt5-x\lt4\)
Subtract 5 from all three parts.
\(\,\,\,\,\,\,-9\lt-x\lt-1\)
Multiply by -1 and reverse both inequality signs.
\(\,\,\,\,\,\,9\gt x\gt1\)
Rewrite in increasing order.
\(\,\,\,\,\,\,1\lt x\lt9\)
\(\,\,\,\,\,\,\left|5-x\right|+1\lt5\)
Subtract 1 from both sides.
\(\,\,\,\,\,\,\left|5-x\right|\lt4\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-4\lt5-x\lt4\)
Subtract 5 from all three parts.
\(\,\,\,\,\,\,-9\lt-x\lt-1\)
Multiply by -1 and reverse both inequality signs.
\(\,\,\,\,\,\,9\gt x\gt1\)
Rewrite in increasing order.
\(\,\,\,\,\,\,1\lt x\lt9\)
\(\textbf{8)}\) \(2\left|1-2x\right|\le10\) The answer is \(-2\le x\le3\)
\(\,\,\,\,\,\,2\left|1-2x\right|\le10\)
Divide both sides by 2.
\(\,\,\,\,\,\,\left|1-2x\right|\le5\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-5\le1-2x\le5\)
Subtract 1 from all three parts.
\(\,\,\,\,\,\,-6\le-2x\le4\)
Divide by -2 and reverse both inequality signs.
\(\,\,\,\,\,\,3\ge x\ge-2\)
Rewrite in increasing order.
\(\,\,\,\,\,\,-2\le x\le3\)
\(\,\,\,\,\,\,2\left|1-2x\right|\le10\)
Divide both sides by 2.
\(\,\,\,\,\,\,\left|1-2x\right|\le5\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-5\le1-2x\le5\)
Subtract 1 from all three parts.
\(\,\,\,\,\,\,-6\le-2x\le4\)
Divide by -2 and reverse both inequality signs.
\(\,\,\,\,\,\,3\ge x\ge-2\)
Rewrite in increasing order.
\(\,\,\,\,\,\,-2\le x\le3\)
\(\textbf{9)}\) \(\left|2x+1\right|\lt-1\) The answer is No Solution.
\(\,\,\,\,\,\,\left|2x+1\right|\lt-1\)
An absolute value is always greater than or equal to zero.
\(\,\,\,\,\,\,\left|2x+1\right|\ge0\)
It can never be less than -1.
Therefore, there is no solution.
\(\,\,\,\,\,\,\left|2x+1\right|\lt-1\)
An absolute value is always greater than or equal to zero.
\(\,\,\,\,\,\,\left|2x+1\right|\ge0\)
It can never be less than -1.
Therefore, there is no solution.
\(\textbf{10)}\) \(\left|1-x\right|\gt1\) The answer is \(x\lt0\text{ or }x\gt2\)
\(\,\,\,\,\,\,\left|1-x\right|\gt1\)
Split into two inequalities.
\(\,\,\,\,\,\,1-x\lt-1\) or \(1-x\gt1\)
Subtract 1 from both sides of each inequality.
\(\,\,\,\,\,\,-x\lt-2\) or \(-x\gt0\)
Divide by -1 and reverse the inequality signs.
\(\,\,\,\,\,\,x\gt2\) or \(x\lt0\)
Rewrite in increasing order.
\(\,\,\,\,\,\,x\lt0\text{ or }x\gt2\)
\(\,\,\,\,\,\,\left|1-x\right|\gt1\)
Split into two inequalities.
\(\,\,\,\,\,\,1-x\lt-1\) or \(1-x\gt1\)
Subtract 1 from both sides of each inequality.
\(\,\,\,\,\,\,-x\lt-2\) or \(-x\gt0\)
Divide by -1 and reverse the inequality signs.
\(\,\,\,\,\,\,x\gt2\) or \(x\lt0\)
Rewrite in increasing order.
\(\,\,\,\,\,\,x\lt0\text{ or }x\gt2\)
\(\textbf{11)}\) \(\left|x-4\right|\lt7\) The answer is \(-3\lt x\lt11\)
\(\,\,\,\,\,\,\left|x-4\right|\lt7\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-7\lt x-4\lt7\)
Add 4 to all three parts.
\(\,\,\,\,\,\,-3\lt x\lt11\)
\(\,\,\,\,\,\,\left|x-4\right|\lt7\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-7\lt x-4\lt7\)
Add 4 to all three parts.
\(\,\,\,\,\,\,-3\lt x\lt11\)
\(\textbf{12)}\) \(\left|3x+2\right|\ge11\) The answer is \(x\le-\frac{13}{3}\) or \(x\ge3\)
\(\,\,\,\,\,\,\left|3x+2\right|\ge11\)
Split into two inequalities.
\(\,\,\,\,\,\,3x+2\le-11\) or \(3x+2\ge11\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,3x\le-13\) or \(3x\ge9\)
Divide both sides by 3.
\(\,\,\,\,\,\,x\le-\frac{13}{3}\) or \(x\ge3\)
\(\,\,\,\,\,\,\left|3x+2\right|\ge11\)
Split into two inequalities.
\(\,\,\,\,\,\,3x+2\le-11\) or \(3x+2\ge11\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,3x\le-13\) or \(3x\ge9\)
Divide both sides by 3.
\(\,\,\,\,\,\,x\le-\frac{13}{3}\) or \(x\ge3\)
\(\textbf{13)}\) \(4\left|x+2\right|-3\gt9\) The answer is \(x\lt-5\) or \(x\gt1\)
\(\,\,\,\,\,\,4\left|x+2\right|-3\gt9\)
Add 3 to both sides.
\(\,\,\,\,\,\,4\left|x+2\right|\gt12\)
Divide both sides by 4.
\(\,\,\,\,\,\,\left|x+2\right|\gt3\)
Split into two inequalities.
\(\,\,\,\,\,\,x+2\lt-3\) or \(x+2\gt3\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,x\lt-5\) or \(x\gt1\)
\(\,\,\,\,\,\,4\left|x+2\right|-3\gt9\)
Add 3 to both sides.
\(\,\,\,\,\,\,4\left|x+2\right|\gt12\)
Divide both sides by 4.
\(\,\,\,\,\,\,\left|x+2\right|\gt3\)
Split into two inequalities.
\(\,\,\,\,\,\,x+2\lt-3\) or \(x+2\gt3\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,x\lt-5\) or \(x\gt1\)
\(\textbf{14)}\) \(2\left|3x-1\right|+5\le17\) The answer is \(-\frac{5}{3}\le x\le\frac{7}{3}\)
\(\,\,\,\,\,\,2\left|3x-1\right|+5\le17\)
Subtract 5 from both sides.
\(\,\,\,\,\,\,2\left|3x-1\right|\le12\)
Divide both sides by 2.
\(\,\,\,\,\,\,\left|3x-1\right|\le6\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-6\le3x-1\le6\)
Add 1 to all three parts.
\(\,\,\,\,\,\,-5\le3x\le7\)
Divide all three parts by 3.
\(\,\,\,\,\,\,-\frac{5}{3}\le x\le\frac{7}{3}\)
\(\,\,\,\,\,\,2\left|3x-1\right|+5\le17\)
Subtract 5 from both sides.
\(\,\,\,\,\,\,2\left|3x-1\right|\le12\)
Divide both sides by 2.
\(\,\,\,\,\,\,\left|3x-1\right|\le6\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-6\le3x-1\le6\)
Add 1 to all three parts.
\(\,\,\,\,\,\,-5\le3x\le7\)
Divide all three parts by 3.
\(\,\,\,\,\,\,-\frac{5}{3}\le x\le\frac{7}{3}\)
\(\textbf{15)}\) \(\left|\frac{x-1}{2}\right|\lt3\) The answer is \(-5\lt x\lt7\)
\(\,\,\,\,\,\,\left|\frac{x-1}{2}\right|\lt3\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-3\lt\frac{x-1}{2}\lt3\)
Multiply all three parts by 2.
\(\,\,\,\,\,\,-6\lt x-1\lt6\)
Add 1 to all three parts.
\(\,\,\,\,\,\,-5\lt x\lt7\)
\(\,\,\,\,\,\,\left|\frac{x-1}{2}\right|\lt3\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-3\lt\frac{x-1}{2}\lt3\)
Multiply all three parts by 2.
\(\,\,\,\,\,\,-6\lt x-1\lt6\)
Add 1 to all three parts.
\(\,\,\,\,\,\,-5\lt x\lt7\)
\(\textbf{16)}\) \(\left|0.5x-2\right|\le3\) The answer is \(-2\le x\le10\)
\(\,\,\,\,\,\,\left|0.5x-2\right|\le3\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-3\le0.5x-2\le3\)
Add 2 to all three parts.
\(\,\,\,\,\,\,-1\le0.5x\le5\)
Multiply all three parts by 2.
\(\,\,\,\,\,\,-2\le x\le10\)
\(\,\,\,\,\,\,\left|0.5x-2\right|\le3\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-3\le0.5x-2\le3\)
Add 2 to all three parts.
\(\,\,\,\,\,\,-1\le0.5x\le5\)
Multiply all three parts by 2.
\(\,\,\,\,\,\,-2\le x\le10\)
\(\textbf{17)}\) \(-2\left|x-5\right|+4\gt-6\) The answer is \(0\lt x\lt10\)
\(\,\,\,\,\,\,-2\left|x-5\right|+4\gt-6\)
Subtract 4 from both sides.
\(\,\,\,\,\,\,-2\left|x-5\right|\gt-10\)
Divide by -2 and reverse the inequality sign.
\(\,\,\,\,\,\,\left|x-5\right|\lt5\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-5\lt x-5\lt5\)
Add 5 to all three parts.
\(\,\,\,\,\,\,0\lt x\lt10\)
\(\,\,\,\,\,\,-2\left|x-5\right|+4\gt-6\)
Subtract 4 from both sides.
\(\,\,\,\,\,\,-2\left|x-5\right|\gt-10\)
Divide by -2 and reverse the inequality sign.
\(\,\,\,\,\,\,\left|x-5\right|\lt5\)
Rewrite as a compound inequality.
\(\,\,\,\,\,\,-5\lt x-5\lt5\)
Add 5 to all three parts.
\(\,\,\,\,\,\,0\lt x\lt10\)
\(\textbf{18)}\) \(\left|2x+6\right|\ge0\) The answer is All Real Numbers.
\(\,\,\,\,\,\,\left|2x+6\right|\ge0\)
The absolute value of any real number is nonnegative.
\(\,\,\,\,\,\,\left|2x+6\right|\ge0\)
This inequality is always true.
Therefore, all real numbers are solutions.
\(\,\,\,\,\,\,\left|2x+6\right|\ge0\)
The absolute value of any real number is nonnegative.
\(\,\,\,\,\,\,\left|2x+6\right|\ge0\)
This inequality is always true.
Therefore, all real numbers are solutions.
\(\textbf{19)}\) \(\left|4x-8\right|\lt0\) The answer is No Solution.
\(\,\,\,\,\,\,\left|4x-8\right|\lt0\)
An absolute value is always greater than or equal to zero.
\(\,\,\,\,\,\,\left|4x-8\right|\ge0\)
It can never be strictly less than zero.
Therefore, there is no solution.
\(\,\,\,\,\,\,\left|4x-8\right|\lt0\)
An absolute value is always greater than or equal to zero.
\(\,\,\,\,\,\,\left|4x-8\right|\ge0\)
It can never be strictly less than zero.
Therefore, there is no solution.
\(\textbf{20)}\) \(5-3\left|2x+1\right|\le-7\) The answer is \(x\le-\frac{5}{2}\) or \(x\ge\frac{3}{2}\)
\(\,\,\,\,\,\,5-3\left|2x+1\right|\le-7\)
Subtract 5 from both sides.
\(\,\,\,\,\,\,-3\left|2x+1\right|\le-12\)
Divide by -3 and reverse the inequality sign.
\(\,\,\,\,\,\,\left|2x+1\right|\ge4\)
Split into two inequalities.
\(\,\,\,\,\,\,2x+1\le-4\) or \(2x+1\ge4\)
Subtract 1 from both sides.
\(\,\,\,\,\,\,2x\le-5\) or \(2x\ge3\)
Divide both sides by 2.
\(\,\,\,\,\,\,x\le-\frac{5}{2}\) or \(x\ge\frac{3}{2}\)
\(\,\,\,\,\,\,5-3\left|2x+1\right|\le-7\)
Subtract 5 from both sides.
\(\,\,\,\,\,\,-3\left|2x+1\right|\le-12\)
Divide by -3 and reverse the inequality sign.
\(\,\,\,\,\,\,\left|2x+1\right|\ge4\)
Split into two inequalities.
\(\,\,\,\,\,\,2x+1\le-4\) or \(2x+1\ge4\)
Subtract 1 from both sides.
\(\,\,\,\,\,\,2x\le-5\) or \(2x\ge3\)
Divide both sides by 2.
\(\,\,\,\,\,\,x\le-\frac{5}{2}\) or \(x\ge\frac{3}{2}\)
See Related Pages\(\)
\(\bullet\text{ Absolute Value Inequality Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Absolute Value Equations}\)
\(\,\,\,\,\,\,\,\,|2x+5|=11…\)
\(\bullet\text{ Andymath Homepage}\)
