Skip to content

Physics Playground

Simulations How we teach Pricing

Language

Log in Start learning free

Loading…

Physics Playground

Built on open, peer-reviewed sources: OpenStax, The Feynman Lectures on Physics, HyperPhysics, NASA and NIST. See our sources.

Learn

Physics course
Math for Physics
Start with the math toolkit
Simulations
Glossary
Exercises
Past exams
Library
How we teach

Company

Pricing
Terms
Privacy
Contact

© 2026 Physics Playground. Made for the curious.Language

Exercises Equations and inequalities

Example 11 from Algebra and Trigonometry, 2.7 Linear Inequalities and Absolute Value Inequalities

Using a Graphical Approach to Solve Absolute Value Inequalities

Given the equation $y = - \frac{1}{2} | 4 x - 5 | + 3,$ determine the x-values for which the y-values are negative.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We are trying to determine where $y < 0,$ which is when $- \frac{1}{2} | 4 x - 5 | + 3 < 0.$ We begin by isolating the absolute value.
$\begin{array}{l} - \frac{1}{2} | 4 x - 5 | < - 3 & Multiply both sides by –2, and reverse the inequality . \\ | 4 x - 5 | > 6 \end{array}$
Next, we solve for the equality $| 4 x - 5 | = 6.$
$\begin{array}{l} 4 x - 5 = 6 & 4 x - 5 = - 6 \\ 4 x = 11 & or & 4 x = - 1 \\ x = \frac{11}{4} & x = - \frac{1}{4} \end{array}$
Now, we can examine the graph to observe where the y-values are negative. We observe where the branches are below the x-axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at $x = - \frac{1}{4}$ and $x = \frac{11}{4},$ and that the graph opens downward. See Figure 5.

Figure 5

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve $- 2 | k - 4 | \leq - 6.$

How did it go?

PreviousSolving an Absolute Value Inequality

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.