The tolerance for the weight of a volleyball is 2.6 grams. If the average volleyball weighs 260 grams, what is the range of weights for a volleyball?
Watch This
Khan Academy: Absolute Value Inequalities
Guidance
Like absolute value equations, absolute value inequalities also will have two answers. However, they will have a range of answers, just like compound inequalities.
Notice in the second inequality, we did not write
Example A
Solve
Solution: There will be two solutions, one with the answer and sign unchanged and the other with the inequality sign flipped and the answer with the opposite sign.
Test a solution,
When graphing this inequality, we have
Notice that this particular absolute value inequality has a solution that is an “and” inequality because the solution is between two numbers.
If
If
If
If
If you are ever confused by the rules above, always test one or two solutions and graph it.
Example B
Solve and graph
Solution: Break apart the absolute value inequality to find the two solutions.
Test a solution,
The graph is:
Example C
Solve
Solution: Given the rules above, this will become an “and” inequality.
The solution is
The graph is:
Intro Problem Revisit Set up an absolute value inequality. w is the range of weights of the volleyball.
So, the range of the weight of a volleyball is
Guided Practice
1. Is
2. Solve and graph
Answers
1. Plug in 4 for
Yes, 4 works, so it is a solution to this absolute value inequality.
2. Split apart the inequality to find the two answers.
Test a solution,
Explore More
Determine if the following numbers are solutions to the given absolute value inequalities.

x−9>4;10 
∣∣∣12x−5∣∣∣≤1;8 
5x+14≥29;−8
Solve and graph the following absolute value inequalities.

x+6>12 
9−x≤16 
2x−7≥3 
8x−5<27 
∣∣∣56x+1∣∣∣>6 
18−4x≤2 
∣∣∣34x−8∣∣∣>13  \begin{align*}67x \le 34\end{align*}
 \begin{align*}19+3x \ge 46\end{align*}
Solve the following absolute value inequalities. \begin{align*}a\end{align*} is greater than zero.
 \begin{align*}xa>a\end{align*}
 \begin{align*}x+a \le a\end{align*}
 \begin{align*}ax \le a\end{align*}