Which of the following gives all the solutions to the... | SAT Nonlinear equations in one variable and systems of equations in two variables Question, Answer Explained

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Question bank / Nonlinear equations in one variable and systems of equations in two variables

Math · Advanced MathMedium

Which of the following gives all the solutions to the equation $|3x - 4| = 11$ ?

$5$ only

$-5$ and $5$

$-\frac{7}{3}$ and $5$

$-5$ and $\frac{7}{3}$

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Answer: C

The expression inside the absolute value can equal $11$ or $-11$ . If $3x - 4 = 11$ , then $3x = 15$ and $x = 5$ . If $3x - 4 = -11$ , then $3x = -7$ and $x = -\frac{7}{3}$ .

Choice B assumes the two solutions are opposites, which isn't true here because of the $-4$ inside the absolute value.

Why the other choices are wrong

A. This uses only 3x - 4 = 11. The case 3x - 4 = -11 gives a second solution.
B. Substituting -5 gives |-19| = 19, not 11. The solutions aren't opposites because of the -4.
D. This has the signs reversed. Substituting -5 gives |-19| = 19, not 11.

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