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16 Sep 2014, 01:49
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Which of the following inequalities has a solution set that, when graphed on a number line, is a single, finite line segment?
A. \(x \ge 4\)
B. \(x^2 \ge 4\)
C. \(x^3 \le 64\)
D. \(|x| \ge 4\)
E. \(|x| \le 4\)
A. \(x \ge 4\)
B. \(x^2 \ge 4\)
C. \(x^3 \le 64\)
D. \(|x| \ge 4\)
E. \(|x| \le 4\)
16 Sep 2014, 01:49
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Bookmarks
Official Solution:
Which of the following inequalities has a solution set that, when graphed on a number line, is a single, finite line segment?
A. \(x \ge 4\)
B. \(x^2 \ge 4\)
C. \(x^3 \le 64\)
D. \(|x| \ge 4\)
E. \(|x| \le 4\)
We must determine which inequality is represented as a single, finite line segment when graphed on a number line.
Choice A: \(x \ge 4\) is represented by a closed dot at \(4\), and a line extending infinitely to the right. Because the line is infinite, eliminate choice A.
Choice B: \(x^2 \ge 4\) has closed dots at 2 and -2, because \(x^2 = 4\) has a positive and a negative root. Taking the square root of both sides yields \(x \ge 2\) and \(x \le -2\), because numbers greater than 2 or less than -2 have squares greater than 4. Thus, one of the solutions of the inequality will be a closed dot at 2 with a line extending infinitely to the right. The other solution will be a closed dot at -2 with a line extending infinitely to the left. This will not be a single line segment. Eliminate choice B.
C: \(x^3 \le 64\) has only one closed dot, at \(x = 4\), because \(x^3\) has only one root. (Values raised to odd powers have one root, while values raised to even powers always have two roots.) We take the cube root of both sides, leaving \(x \le 4\), which is represented by a closed dot at \(4\) with a line extending infinitely to the left. We can eliminate choice C.
Choice D: Because we are dealing with absolute value, we must consider the case when \(x\) is positive AND the case when \(x\) is negative; both yield the same absolute value: \(|x| = |-x|\). Thus, either \(x \ge 4\), or \(-x \ge 4\). We divide both sides of \(-x \ge 4\) by -1, remembering to flip the sign. This gives us: \(x \le -4\). The inequality in Choice D is thus represented by two regions: a closed dot at 4 with a line extending infinitely to the right, and a closed dot at -4 with a line extending infinitely to the left. Eliminate choice D.
Because the first four answer choices are wrong, we know that the correct answer must be choice E. We can check to make sure:
Choice E: As with choice D, we must consider the case when \(x\) is positive and the case when \(x\) is negative, since both yield the same absolute value. Thus, either \(x \le 4\) or \(-x \le 4\). Multiplying this second inequality by -1 gives \(x \ge -4\), for a combined inequality of: \(-4 \le x \le 4\). This inequality is represented with a single, finite region: a closed dot at 4 with a line extending to the left to a closed dot at -4.
Answer: E
Which of the following inequalities has a solution set that, when graphed on a number line, is a single, finite line segment?
A. \(x \ge 4\)
B. \(x^2 \ge 4\)
C. \(x^3 \le 64\)
D. \(|x| \ge 4\)
E. \(|x| \le 4\)
We must determine which inequality is represented as a single, finite line segment when graphed on a number line.
Choice A: \(x \ge 4\) is represented by a closed dot at \(4\), and a line extending infinitely to the right. Because the line is infinite, eliminate choice A.
Choice B: \(x^2 \ge 4\) has closed dots at 2 and -2, because \(x^2 = 4\) has a positive and a negative root. Taking the square root of both sides yields \(x \ge 2\) and \(x \le -2\), because numbers greater than 2 or less than -2 have squares greater than 4. Thus, one of the solutions of the inequality will be a closed dot at 2 with a line extending infinitely to the right. The other solution will be a closed dot at -2 with a line extending infinitely to the left. This will not be a single line segment. Eliminate choice B.
C: \(x^3 \le 64\) has only one closed dot, at \(x = 4\), because \(x^3\) has only one root. (Values raised to odd powers have one root, while values raised to even powers always have two roots.) We take the cube root of both sides, leaving \(x \le 4\), which is represented by a closed dot at \(4\) with a line extending infinitely to the left. We can eliminate choice C.
Choice D: Because we are dealing with absolute value, we must consider the case when \(x\) is positive AND the case when \(x\) is negative; both yield the same absolute value: \(|x| = |-x|\). Thus, either \(x \ge 4\), or \(-x \ge 4\). We divide both sides of \(-x \ge 4\) by -1, remembering to flip the sign. This gives us: \(x \le -4\). The inequality in Choice D is thus represented by two regions: a closed dot at 4 with a line extending infinitely to the right, and a closed dot at -4 with a line extending infinitely to the left. Eliminate choice D.
Because the first four answer choices are wrong, we know that the correct answer must be choice E. We can check to make sure:
Choice E: As with choice D, we must consider the case when \(x\) is positive and the case when \(x\) is negative, since both yield the same absolute value. Thus, either \(x \le 4\) or \(-x \le 4\). Multiplying this second inequality by -1 gives \(x \ge -4\), for a combined inequality of: \(-4 \le x \le 4\). This inequality is represented with a single, finite region: a closed dot at 4 with a line extending to the left to a closed dot at -4.
Answer: E
