GMAT Quantitative - Absolute Value

Solve $\left | 2x - 5 \right |\geq 3$ .

$x \leq 1, x\geq 4$

$x \leq -1, x\geq -4$

$1 < x < 4$

$-2 \leq x\leq 5$

$x < 1, x > 4$

Explanation

It's actually easier to solve for the complement first. Let's solve $\left | 2x-5 \right |<3$ . That gives -3 < 2x - 5 < 3. Add 5 to get 2 < 2x < 8, and divide by 2 to get 1 < x < 4. To find the real solution then, we take the opposites of the two inequality signs. Then our answer becomes $x\leq 1 \textsc{ or } x\geq 4$ .

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

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

or $\frac{1}{3}$

$x=-\frac{1}{3}$

Explanation

$\left | 3x - 7 \right |=8$ 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 $\dpi{100} -\frac{1}{3}$ .

Simplify the following expression:

$\small -34$

$\small 34$

$\small \small -2$

$\small -2$

$\small 170$

Explanation

This question plays a few tricks dealing with absolute values. To begin, we can recognize that any negative sign within an absolute value can basically be rendered positive. So this:

becomes:

$\small 34*2*-\left | 0.5\right |$

In this case, we still have a negative that was outside of the absolute value sign. This term will stay negative, so we get:

$\small \small 34*2*- 0.5=-34$

This makes our answer $\small -34$ .

Solve the following inequality:

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

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

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

Explanation

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 :

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

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

Solve for :

$\left | 6x-4\right |=14$

and $x=-\frac{5}{3}$

$x=-\frac{5}{3}$

and $x=\frac{5}{3}$

Not enough information to solve

Explanation

In order to solve the given absolute value equation, we need to solve for for the two ways in which this absolute value can be solved:

1.)

2.)

Solving Equation 1:

Solving Equation 2:

$x=-\frac{10}{6}=-\frac{5}{3}$

Therefore, there are two solutions to the absolute value equation: and $x=-\frac{5}{3}$

Explanation

Remember that the absolute value of any number is its positive value, regardless of whether or not the number is negative before the absolute value is taken. We start by simplifying any expressions inside the absolute value signs: