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Chapter 2: Problem 21

Solve the inequality. Then graph the solution set. $$x^{2}+2 x-3<0$$

Short Answer

Expert verified
The solution set to the inequality is \(-3 < x < 1\), and the graphical representation will be a number line spanning from -3 to 1, excluding these points.

Step by step solution

01

Factoring the inequality

Rewrite the given inequality, \(x^{2} + 2x - 3 < 0\), in a factored form. By making use of the factoring technique, specifically the factoring of quadratic trinomials, we have \((x - 1)(x + 3) < 0\).
02

Identifying the critical points

The critical points or roots of the inequality \(x^{2} + 2x - 3 < 0\) are obtained by setting each factor to zero and solving for \(x\). Doing so gives us \(x_1 = 1\) and \(x_2 = -3\). These represent the boundary points of the intervals to test.
03

Testing the intervals

Choose a test number within each interval. Keep in mind that solutions will not include the critical points themselves, as they make the expression equal to zero, not less than zero. Thus, the test numbers might be \(x=-4\), \(x=0\), and \(x=2\). Substitute each test number into the factored inequality to decide which intervals are part of the solution set. The factored inequality will either provide a positive or negative sign, associate each interval with the sign, and select only the intervals providing a negative sign.
04

Drawing the inequality graph

Draw a number line spanning the critical points and include the intervals contributing to a negative sign from step 3, while excluding the critical points because the inequality is \(< 0\) and not \(\leq 0\).

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Most popular questions from this chapter

Describe the error. $$\sqrt{-6} \sqrt{-6}=\sqrt{(-6)(-6)}=\sqrt{36}=6$$

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