# Algebra 1 : How to solve absolute value equations

## Example Questions

### Example Question #51 : How To Solve Absolute Value Equations

Solve for

or

or

or

Explanation:

We define the absolute value of a number as that number's distance from  on the number line. Given , we therefore must solve for two possibilities:

1.) , or

2.)

Solving #1, we get:

Solving #2, we get:

Therefore, the solution is  or .

### Example Question #52 : How To Solve Absolute Value Equations

Solve for

Does Not Exist

Does Not Exist

Explanation:

We define the absolute value of a number as that number's distance from  on the number line.

Given

,

we know that the absolute value cannot exist because distance can never be described as a negative number.

### Example Question #53 : How To Solve Absolute Value Equations

Explanation:

Before we can solve for x, we must isolate the absolute value symbol on one side of the equation. To do that, we must subtract  from both sides, leaving us with

Next, we must make two equations:  and .

From here, it is a simple one-step equation, in which we divide both sides by . This gives us

.

### Example Question #54 : How To Solve Absolute Value Equations

Solve for

or

or

Explanation:

We define the absolute value of a number as that number's distance from on the number line.

Given , we therefore must solve for two possibilities:

1.) , or

2.)

Solving #1, we get:

Solving #2, we get:

Consequently, the solution is  or .

### Example Question #55 : How To Solve Absolute Value Equations

Solve for

or

or

or

Explanation:

We define the absolute value of a number as that number's distance from  on the number line.

Given , we therefore must solve for two possibilities:

1.) , or

2.)

Solving #1, we get:

Solving #2, we get:

Consequently, the solution is  or .

### Example Question #56 : How To Solve Absolute Value Equations

Solve for

or

or

or

Explanation:

We define the absolute value of a number as that number's distance from  on the number line.

Given , or  we therefore must solve for two possibilities:

1.) , or

2.)

Solving #1, we get:

Solving #2, we get:

Consequently, the solution is  or  .

### Example Question #891 : Linear Equations

Solve for all values of x:

Explanation:

The first step is to split the absolute value into two equations, one case for when the inside of the absolute value will be negative and one case for when the inside of the absolute value will be positive.

Positive case:

First subtracting 2 and 4x from both sides:

dividing both sides by -2:

Negative case:

First move the negative to the other side:

Next, add 4x and subtract 2 from both sides:

Finally, divide both sides by 6 and reducing:

Thus the solutions are:

### Example Question #892 : Linear Equations

and

and

and

and

Explanation:

Absolute value symbols tell you to take whatever number is inside the absolute value and make it positive. This is important to remember for equations with variables like x because if the variable is inside the absolute value symbols, that means two values for the variable will give the right answer. You solve for both the positive and the negative of whatever number is on the other side of the equation, since the absolute value symbols will turn both positive and negative numbers into positive ones. For example, in this problem, the absolute value of 2x+4 is alone on one side of the equation with a variable (x) in it. Set 2x+4 equal to both 16 and -16 in two different equations removing the absolute value sign.

You should have  and . Solve to get x alone.

Subtract 4 from both sides of the equation to get

and

Divide by 2 on both sides of the equation to get

and

If you want to check these answers plug them back into the original equation.

Now try the other solution.

Both solutions for x check out to give the answer 16.

### Example Question #893 : Linear Equations

and

and

No solution

and

Explanation:

This problem should be easy if you have a basic understanding of absolute values. Knowing that absolute values make whatever numbers inside them positive, ask yourself, what values for x will equal positive or negative 9? You can again set up two equations.

and

These two equations are already solved for x which means we have our answers.

Double checking by plugging these back into the original equation...

Plugging in 9.

Now to try plugging in -9.

### Example Question #51 : How To Solve Absolute Value Equations

and

and

No solution

No solution

Explanation:

This question is a bit of a trick. Absolute values always make the numbers in them positive. If an absolute value is left alone on one side of the equation and the other side of the equation is a negative number then there cannot be a possible solution.

Here is where checking your answer is important! Say you did not see the negative on the right side of the equation and thought you were supposed to solve the absolute value for both the positive and negative value of the other side

Remember this is the incorrect way to do this problem!

and

and

Plugging these answers back into the equation you can see why this is wrong.

and

Neither of these answers is right. There is no solution.