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Chapter 2: Problem 20

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{2}+2 x<0$$

Short Answer

Expert verified
The solution to the inequality \(x^{2}+2 x<0\) is (-2, 0).

Step by step solution

01

Rearrange and Factor

Rewrite the inequality \(x^{2}+2 x<0\) as \(x(x+2)<0\). This is a factored form.
02

Identify Critical Values

Set each part of the inequality which has been factored to zero to find the critical values. These are solutions to the equation \(x(x+2)=0\), which yields \(x=0\) and \(x=-2\). These are the boundaries for intervals to test.
03

Test Intervals

The critical values split the number line into intervals, (-infinity, -2), (-2, 0), and (0, infinity). Pick a test value in each interval and substitute into the factored inequality: for the interval (-infinity, -2) choose x = -3, for (-2, 0) choose x = -1, and for (0, infinity) choose x = 1. We seek instances where the inequality is less than zero (i.e., negative). The intervals that satisfy this condition are (-2, 0) as \(x(x+2)\) gives a negative result for x = -1 and other values in the interval. When testing values from other intervals, we see that the expression evaluates to a positive value.
04

Express the Solution in Interval Notation

The solution to the inequality \(x^{2}+2 x<0\) on interval notation is (-2, 0). This means that the solution set is all values between -2 and 0, not inclusive of -2 and 0 since our original problem is a 'less than' and not a 'less than or equal to' inequality.

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