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.
\(\textbf{1)}\) \( \left|5-2x\right|\le10 \)
\(\textbf{2)}\) \( 2\left|2x-3\right|\gt10 \)
\(\textbf{3)}\) \( 3\left| x+1\right|+20\le5 \)
\(\textbf{4)}\) \( 3\left| x\right|+4\ge-8 \)
\(\textbf{5)}\) \( 3\left| 2x-3\right|-5\le10 \)
\(\textbf{6)}\) \( \left| 3x-7\right|\le0 \)
\(\textbf{7)}\) \(\left| 5 – x \right|+1 \lt 5\)
\(\textbf{8)}\) \(2\left| 1 – 2x \right| \le 10\)
\(\textbf{9)}\) \(\left| 2x + 1 \right| \lt – 1\)
\(\textbf{10)}\) \(\left| 1 – x \right| \gt 1\)
See Related Pages\(\)
\(\bullet\text{ Absolute Value Inequality Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Absolute Value Equations}\)
\(\,\,\,\,\,\,\,\,|2x+5|=11…\)
\(\bullet\text{ Andymath Homepage}\)
About Andymath.com
Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.
Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.
Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!
Thank you for visiting. How exciting!