GMAT Math : Understanding absolute value

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Example Questions

Example Question #465 : Algebra

$\displaystyle f(x) = |x+ 5|$

$\displaystyle g(x) = |11 - x|$

If $(f \circ g )(x) = 8$ $\displaystyle (f \circ g )(x) = 8$, then how many possible values of $\displaystyle x$ are there?

Possible Answers:

Four

Zero

Two

One

Three

Correct answer:

Two

Explanation:

$(f \circ g )(x) = 8$ $\displaystyle (f \circ g )(x) = 8$

can be rewritten as

$\displaystyle f [ g (x) ]= 8$

$\displaystyle f(x) = |x+ 5|$, so

$\displaystyle f [ g (x) ] = | g(x)+ 5|$.

If $(f \circ g )(x) = 8$ $\displaystyle (f \circ g )(x) = 8$, then

$\displaystyle | g(x)+ 5| = 8$, or, equivalently, either

$\displaystyle g(x)+ 5 = -8$ or $\displaystyle g(x)+ 5 = 8$.

Solve separately:

$\displaystyle g(x)+ 5 = -8$

$\displaystyle g(x) = -13$

$\displaystyle g(x)+ 5 = 8$

g(x)= 3 $\displaystyle g(x)= 3$

$\displaystyle g(x) = |11 - x|$, so the above two statements can be rewritten as

$\displaystyle |11 - x| = -13$ and $\displaystyle |11 - x| = 3$

$\displaystyle |11 - x| = -13$ has no solution, since the absolute value of a number cannot be negative.

$\displaystyle |11 - x| = 3$ can be rewritten as

$\displaystyle 11 - x = 3$ and $\displaystyle 11 - x = -3$

It is not necessary to solve these statements, as we can determine that the correct response is two solutions.

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Example Question #31 : Absolute Value

Solve for $\displaystyle x$:

$\left | 3x + 18\right | = 9$ $\displaystyle \left | 3x + 18\right | = 9$

Possible Answers:

x = 9 $\displaystyle x = 9$

x = -3 $\displaystyle x = -3$

$\displaystyle x = -3, -9$

$\displaystyle x= -3, 9$

x= 3, 9 $\displaystyle x= 3, 9$

Correct answer:

$\displaystyle x = -3, -9$

Explanation:

To solve absolute value equations, we must set up two equations: one where the solution is negative, and one where the solution is positive.

$\left | 3x + 18\right | = 9$ $\displaystyle \left | 3x + 18\right | = 9$

$\displaystyle 3x + 18 = 9$ $\displaystyle 3x + 18 = -9$

3x = -9 $\displaystyle 3x = -9$ $\displaystyle 3x = -27$

x = -3 $\displaystyle x = -3$ x = -9 $\displaystyle x = -9$

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Example Question #467 : Algebra

True or false: $\displaystyle X$ is a positive number.

Statement 1: $\displaystyle |X - 21| < 5$

Statement 2: $\displaystyle |X - 20| > 3$

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone.

$\displaystyle |X - 21| < 5$ can be rewritten as

$\displaystyle -5 < X - 21 < 5$

$\displaystyle -5 + 21 < X - 21 + 21< 5 + 21$

$\displaystyle 16 < X < 26$

Therefore, $\displaystyle X$ is positive.

Assume Statement 2 alone. The sign of $\displaystyle X$ cannot be determined. For example, if X = 24 $\displaystyle X = 24$, which is positive, then

$\displaystyle |X - 20| = |24 - 20| = |4| = 4 > 3$.

If X = -1 $\displaystyle X = -1$, which is not positive, then

$\displaystyle |X - 20| = |-1 - 20| = |-21| = 21 > 3$.

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Example Question #461 : Algebra

$\displaystyle f(x) = |10-x|$

$\displaystyle g(x) = |x-10|$

How many values of $\displaystyle x$ make

$(f \circ g )(x) = 0$ $\displaystyle (f \circ g )(x) = 0$

a true statement?

Possible Answers:

None

Two

One

Four

Three

Correct answer:

Two

Explanation:

$(f \circ g )(x) = f[g(x)]$ $\displaystyle (f \circ g )(x) = f[g(x)]$, so we want the number of values of $\displaystyle x$ for which

$\displaystyle f[g(x)] = 0$.

$\displaystyle f(x) = |10-x|$, so

$\displaystyle f[g(x)] = |10-g(x)|$

Therefore, if $\displaystyle f[g(x)] = 0$, then

$\displaystyle |10-g(x)| = 0$

$\displaystyle 10 - g(x) = 0$

$\displaystyle g(x) = 10$

$\displaystyle |x-10| = 10$

Either

$\displaystyle x - 10 = 10$, in which case x =20 $\displaystyle x =20$, or

$\displaystyle x - 10 = -10$, in which case x = 0 $\displaystyle x = 0$.

The correct choice is therefore two.

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Example Question #469 : Algebra

$\displaystyle f(x) = |x+ 6|$

$\displaystyle g(x) = |x-6|$

How many values of $\displaystyle x$ make

$(f \circ g )(x) = 3$ $\displaystyle (f \circ g )(x) = 3$

a true statement?

Possible Answers:

Three

None

Four

Two

One

Correct answer:

None

Explanation:

$(f \circ g )(x) = f[g(x)]$ $\displaystyle (f \circ g )(x) = f[g(x)]$, so we want the number of values of $\displaystyle x$ for which

$\displaystyle f[g(x)] = 3$

$\displaystyle f(x) = |x+ 6|$

$\displaystyle f[g(x)] = |g(x)+ 6| = 3$, so either

$\displaystyle g(x)+ 6 = -3$ or $\displaystyle g(x)+ 6 = 3$

If the first equation is true, then

$\displaystyle g(x)+ 6 = -3$

$\displaystyle g(x)= -9$

and

$\displaystyle |x-6| = -9$.

If the second equation is true, then

$\displaystyle g(x)+ 6 = 3$

$\displaystyle g(x)= -3$

and

$\displaystyle |x-6| = -3$.

In each situation, the absolute value of an expression would be negative; since the absolute value of an expression cannot be negative, no solution is yielded.

There are no values of $\displaystyle x$ that make $(f \circ g )(x) = 3$ $\displaystyle (f \circ g )(x) = 3$ true; the correct response is zero.

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GMAT Math : Understanding absolute value

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #465 : Algebra

Example Question #31 : Absolute Value

Example Question #467 : Algebra

Example Question #461 : Algebra

Example Question #469 : Algebra

All GMAT Math Resources

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