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Chapter 2: Problem 59

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{x-2}{x+2} \leq 2$$

Short Answer

Expert verified
The solution set of the inequality \(\frac{x - 2}{x + 2} \leq 2\) is expressed in interval notation as (-鈭�, -2] 鈭� [2, 鈭�).

Step by step solution

01

Simplify the Inequality

Solve the inequality by subtracting 2 from both sides to set the inequality to 0. This transforms the inequality into \(\frac{x - 2}{x + 2} - 2 \leq 0\). Multiply it out and simplify to get \(x^2 - 4x - 4 \leq 0\).
02

Set the Expression Equal to Zero

Set \(x^2 - 4x - 4 = 0\) to find the boundary points of the intervals to test. Solving this through factoring or the quadratic formula reveals roots -2 and 2.
03

Test Each Interval

Plug in test points from each of the intervals (-鈭�, -2), (-2, 2), and (2, 鈭�) into the inequality to determine whether those intervals are included in the solution set. For instance, if we pick -3, 0 and 3 for each of the intervals respectively, we find that -3 and 3 satisfy the inequality, but 0 does not.
04

Determine the Solution Set

The suitable testing in the intervals leads to the solution set (-鈭�, -2] and [2, 鈭�).
05

Express the Solution Set in Interval Notation

This step doesn't require any computation, it's more related to correct notation. The solution set in interval notation is (-鈭�, -2] 鈭� [2, 鈭�). This means x is less or equal to -2 or x is larger or equal to 2.

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