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Exercises Equations and inequalities

Example 7 from Algebra and Trigonometry, 2.7 Linear Inequalities and Absolute Value Inequalities

Solving a Compound Inequality

Solve the compound inequality: $3 \leq 2 x + 2 < 6.$

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
The first method is to write two separate inequalities: $3 \leq 2 x + 2$ and $2 x + 2 < 6.$ We solve them independently.

$\begin{array}{l} 3 \leq 2 x + 2 & and & 2 x + 2 < 6 \\ 1 \leq 2 x & 2 x < 4 \\ \frac{1}{2} \leq x & x < 2 \end{array}$
Then, we can rewrite the solution as a compound inequality, the same way the problem began.

$\frac{1}{2} \leq x < 2$

In interval notation, the solution is written as $[\frac{1}{2}, 2) .$

The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.

$\begin{array}{l} 3 \leq 2 x + 2 < 6 \\ 1 \leq 2 x < 4 & Isolate the variable term, and subtract 2 from all three parts . \\ \frac{1}{2} \leq x < 2 & Divide through all three parts by 2 . \end{array}$
We get the same solution: $[\frac{1}{2}, 2) .$

Your turn

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

Solve the compound inequality: $4 < 2 x - 8 \leq 10.$

How did it go?

PreviousSolving an Inequality with Fractions Next Solving a Compound Inequality with the Variable in All Three Parts

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.