Absolute Value - GMAT Quantitative

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 - 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 = 15Rightarrow x = 5

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

So or .

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 xleq 1 textsc{ or } xgeq 4.

Since , we know the following:

;

;

;

;

.

Also, we need to compare absolute values, so the greatest one must be either or .

We also know that when .

Thus, we know for sure that .

Set and solve for :

Rewrite as a compound equation and solve each part separately:

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

This is the only number that fits the criterion.

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

If is nonnegative, then , and we can rewrite this as

Solve:

If is negative, then , and we can rewrite this as

The numbers have the given characteristics.

The correct answer is that there is no .

We start by adding to both sides giving

Then multiply both sides by .

Then divide both sides by

Now it is impossible to go any further. The absolute value of any quantity is always positive (or sometimes ). Here we have the absolute value of something equaling a negative number. That's never possible, hence there is no that makes this a true equation.

We start by isolating the expression with the absolute value:

becomes

So: or

We then solve the two equations above, which gives us 42 and 4 respectively.

So the solution is

To solve an inequality we need to remember what the absolute value sign says about our expression. In this case it says that

can be written as

Of .

Rewriting this in one inequality we get:

From here we add one half to both sides .

Finally, we divide by two to isolate and solve for m.

Only is between -1.75 and 2.25

We proceed as follows

(Start)

(Subtract 3 from both sides)

or (Quantity inside the absolute value can be positive or negative)

or (add five to both sides)

or

Another way to say this is

There are two ways to solve the absolute value portion of this problem:

or

From here, you can solve each of these equations independently to arrive at the correct answer:

or

or

The solution is .

This is a problem where we need to use our translating skills. We are given a word problem and asked to solve it. To do so, we need to rewrite our word problem as an equation and then use arithmetic to find the answer. In these types of problems, the hardest step is usually translating correctly, so make sure to be meticulous and work step-by-step!

1)"The absolute value of negative seventeen": Recall that absolute value means that we will just change the sign to positive. Missing that will end up giving you the trap answer .

2)"is multiplied by a number which is three fewer than twelve." We need a number that is three fewer than twelve, so we need to subtract. Follow it up with multiplication and you get:

3)"The resulting number is subtracted fromnegative six." The key word here is "from"—make sure you aren't computing , which would result in another one of the trap answers!

The correct answer is .

Since we are solving an absolute value equation, , we must solve for both potential values of the equation:

1.)

2.)

Solving Equation 1:

Solving Equation 2:

Therefore, for , or .

Since we are solving an absolute value equation, , we must solve for both potential values of the equation:

1.)

2.)

Solving Equation 1:

Solving Equation 2:

Therefore, for , or .

An absolute value of a number must always assume a nonnegative value, so

, and

Therefore,

and the range of is the set .