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Chapter 2: Problem 100

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

Short Answer

Expert verified
The statement is false. The solution set of \( (x+3)(x-1) \geq 0 \) is (-3, 1), while the solution set of \(\frac{x+3}{x-1}\geq 0\) is (-3, 1) union (1, 鈭�). Thus, changing the statement to '\((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} > 0\) have the same solution set' makes it true.

Step by step solution

01

Solve the first inequality

The inequality \((x+3)(x-1) \geq 0\) is a factored quadratic inequality. To find the solution set, first, set each factor to 0 and solve for x to find the critical points: \(x+3=0\), so \(x=-3\) and \(x-1=0\), so \(x=1\). Then test a number from each interval determined by the critical points in the original inequality. For example, select -4, 0, and 2. Writing the intervals, and their corresponding truth values we find: (-鈭�, -3), false; (-3,1), true; and (1, 鈭�), false.
02

Solve the second inequality

The inequality \(\frac{x+3}{x-1}\geq 0\) is a rational inequality. First, set the numerator and denominator to 0 and solve for x to get the critical points: \(x+3=0\), so \(x=-3\) and \(x-1=0\), so \(x=1\). Note that x = 1, which would make the denominator 0, is excluded from the domain of the rational inequality. If we test the intervals using the same test points -4, 0, and 2, we find: (-鈭�, -3), false; (-3,1), true; and (1, 鈭�), true.
03

Compare the solutions

Comparing the truth values for each interval of the two inequalities, we see that they are not the same, thus the solution sets of the two inequalities are not the same.

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