# SAT MATH - ABSOLUTE VALUE EQUATIONS

The absolute value of a number is the distance on a number line between the graph of the number and the origin.

The distance between -3 and the origin is 3. Thus,

|-3| = 3

The distance between 3 and the origin is 3. Thus,

|3| = 3

Therefore, if |x| = 3, then x = 3 or x = -3.

An absolute value function can be written using two or more expressions such as

## How to Solve Absolute Value Equations

To solve any absolute value function, it has to be in the form of

|x + a| = k

Here, a and k are real numbers. And there should be only absolute part on the left side.

Let us consider the absolute value equation given below.

|2x + 3| = 5

The following steps will be useful to solve absolute value equations.

Step 1 :

Get rid of absolute sign and split up into two branches.

Step 2 :

For the first branch, take the sign as it is on the right side.

Step 3 :

For the second branch, change the sign on the right side.

Step 4 :

Then solve both the branches.

## How to Graph Absolute Value Functions

To sketch the graph of y = a|x + h| + k, use the following steps.

1. Find the x-coordinate of the vertex by finding the value of x for which x + h = 0.

2. Make a table of values using the -xcoordinate of the vertex. Find two -xvalues to its left and two to its right.

3. Plot the points from inside the table. If a > 0, the vertex will be the minimum point and if a < 0, the vertex will be the maximum point.

Table of values :

The graph of y = |x|.

## Solved Problems

Problems 1-3 : Solve the absolute value equation.

Problem 1 :

|3x - 5| = 7

Solution :

3x - 5 = 7  or  3x - 5 = -7

3x = 12  or  3x = -2

x = 4  or  x = -

Problem 2 :

|x + 5| = 0

Solution :

|x + 5| = 0

x + 5 = 0

x = -5

Problem 3 :

|x - 2| = -5

Solution :

|x - 2| = -5 means that the distance between x and 2 is -5. Since distance cannot be negative, the given absolute value equation has no solution.

Problem 4 :

Which of the following expressions is equal to -1 for some values of x?

(A)  |1 - x| + 6

(B)  |1 - x| + 4

(C)  |1 - x| + 2

(D)  |1 - x| - 2

Solution :

By definition, the absolute value of any expression is a nonnegative number.

In options (A), (B) and (C),

|1 - x| + 6 > 0

|1 - x| + 4 > 0

|1 - x| + 2 > 0

So, the expressions in options (A), (B) and (C) can not be equal to -1.

In option (D), |1 - x| - 2 could be a negative value.

|1 - x| - 2 = -1

|1 - x| = 1

1 - x = 1  or  1 - x = -1

-x = 0  or  -x = -2

x = 0  or  x = 2

Therefore, the correct answer is option (D).

Problem 5 :

If |2x + 7|= 5, which of the following could be the value of x?

(A)  -6

(B)  -4

(C)  -2

(D)  0

Solution :

|2x + 7|= 5

2x + 7 = 5  or  2x + 7 = -5

2x = -2  or  2x = -12

x = -1  or  x = -6

Therefore, the correct answer is option (A).

Problem 6 :

For what value of x is |x - 1| - 1 equal to 1?

(A)  -1

(B)  0

(C)  1

(D)  2

Solution :

|x - 1| - 1 = 1

x - 1 = 1  or  x - 1 = -1

x = 2  or  x = 0

Therefore, the correct answer is option (B).

Problem 7 :

For what value of x is |3x - 5| equal to -1?

(A)  -2

(B)  -1

(C)  0

(D)  There is no such value of x.

Solution :

Given :

|3x - 5| = -1

Absolute value of any expression can never be negative. So, the equation |3x - 5| = -1 has no solution.

Therefore, the correct answer is option (D).

Problem 8 :

For what value of x is 2|x + 5| - 8 equal to 0?

Solution :

2|x + 5| - 8 = 0

2|x + 5| = 8

|x + 5| = 4

x + 5 = 4  or  x + 5 = -4

x = -1  or  x = -9

Problem 9 :

For what value of x is 3 - |3 - x| equal to 3?

Solution :

3 - |3 - x| = 3

-|3 - x| = 0

|3 - x| = 0

3 - x = 0

x = 3

Problem 10 :

The graph of the function f is shown in the xy-plane above. For what value of x is the value of f(x) at its maximum?

(A)  -3

(B)  -1

(C)  1

(D)  3.

Solution :

The maximum value of the function corresponds to the y- coordinate of the point on the graph, which is highest along the vertical axis. The highest point along the y- axis has coordinates (1, 4).

When f(x) is maximum, the value of x is 1.

Therefore, the correct answer is option (C).

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