Absolute Value - GMAT Quantitative

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Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

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Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

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Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If $x \ge 10$ , since both and are nonnegative, we can rewrite as

, or

.

On $[10, \infty)$ , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is $[f(10), \infty)$ , or, since

,

$[4, \infty)$ .

If , since is negative and is positive, we can rewrite as

, or

is a constant function on this interval and its range is $\{4 \}$ .

If $x \le 6$ , since both and are nonpositive, we can rewrite as

, or

.

On $[- \infty, 6)$ , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is $[f(6), \infty)$ , or, since

, $[4, \infty)$ .

The union of the ranges is the range of the function - $[4, \infty)$ .

← Didn't Know|Knew It →

Question

Give the range of the function

Tap to reveal answer

Answer

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both and are positive, we can rewrite as

, or

,

a constant function with range $\left \{ -4\right \}$ .

If $6 \le x \le 10$ , since is negative and is positive, we can rewrite as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both and are negative, we can rewrite as

, or

,

a constant function with range $\left \{ 4\right \}$ .

The union of the ranges is the range of the function - - which is not among the choices.

← Didn't Know|Knew It →

Question

Give the -intercept(s), if any, of the graph of the function $f (x) = \left | 4x + B \right | - 8$ in terms of .

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Answer

Set and solve for :

$\left | 4x + B \right | - 8 = 0$

$\left | 4x + B \right | = 8$

Rewrite as a compound equation and solve each part separately:

$4x + B = -8 \textrm{ or } 4x + B = 8$

$4x \div 4 =\left ( -B -8 \right ) \div 4$

$x = \frac{-B -8 }{4}$

$4x \div 4 =\left ( -B +8 \right ) \div 4$

$x = \frac{-B +8 }{4}$

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Question

A number is ten less than its own absolute value. What is this number?

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Answer

We can rewrite this as an equation, where is the number in question:

A nonnegative number is equal to its own absolute value, so if this number exists, it must be negative.

In thsi case, , and we can rewrite that equation as

$2N \div 2 = -10\div 2$

This is the only number that fits the criterion.

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Question

Solve the following inequality:

$\left | 2x+3\right |<6$

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Answer

To solve this absolute value inequality, we must remember that the absolute value of a function that is less than a certain number must be greater than the negative of that number. Using this knowledge, we write the inequality as follows, and then perform some algebra to solve for :

$\left | 2x+3\right |<6$

$-\frac{9}{2}<x<\frac{3}{2}$

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Question

Solve $\left | 3x - 7 \right |=8$ .

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Answer

$\left | 3x - 7 \right |=8$ really consists of two equations: $3x - 7 = \pm 8$

We must solve them both to find two possible solutions.

$3x - 7 = 8 \Rightarrow 3x = 15\Rightarrow x = 5$

$3x - 7 = - 8 \Rightarrow 3x = -1\Rightarrow x = -1/3$

So or $\dpi{100} -\frac{1}{3}$ .

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