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Exercises Equations and inequalities

Example 9 from Algebra and Trigonometry, 2.7 Linear Inequalities and Absolute Value Inequalities

Determining a Number within a Prescribed Distance

Describe all values $x$ within a distance of 4 from the number 5.

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 want the distance between $x$ and 5 to be less than or equal to 4. We can draw a number line, such as in Figure 4, to represent the condition to be satisfied.

Figure 4

The distance from $x$ to 5 can be represented using an absolute value symbol, $| x - 5 | .$ Write the values of $x$ that satisfy the condition as an absolute value inequality.
$| x - 5 | \leq 4$

We need to write two inequalities as there are always two solutions to an absolute value equation.

$\begin{array}{l} x - 5 \leq 4 & and & x - 5 \geq - 4 \\ x \leq 9 & x \geq 1 \end{array}$
If the solution set is $x \leq 9$ and $x \geq 1,$ then the solution set is an interval including all real numbers between and including 1 and 9.

So $| x - 5 | \leq 4$ is equivalent to $[1, 9]$ in interval notation.

Your turn

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

Describe all x-values within a distance of 3 from the number 2.

How did it go?

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