How to solve absolute value equations - Algebra

When dealing with absolute value, we need to consider positive and negative values.

Therefore, we will create two separate equations to solve

and .

For the second equation divide both sides by to get .

Thus, our solutions for are,

.

First, split into two possible scenarios according to the absolute value.

Looking at , we can solve for x by subtracting 3 from both sides, so that we get x = 1.

Looking at , we can solve for x by subtracting 3 from both sides, so that we get x = –7.

So therefore, the solution is x = –7, 1.

|x – 3| = 2 means that it can be separated into x – 3 = 2 and x – 3 = –2.

So both x = 5 and x = 1 work.

x – 3 = 2 Add 3 to both sides to get x = 5

x – 3 = –2 Add 3 to both sides to get x = 1

Because of the absolute value signs,

or

Subtract 2 from both sides of both equations:

or

or

There are two answers to this problem:

and

An absolute value expression differs from a normal expression only in its sign. Instead of being a positive or negative quantity, an absolute value represents a scalar distance from zero, so it does not have a sign. For example, is the same as because both represent a value 2 units away from zero. In this problem, equals , or 5. equals 8. The final answer is or 40.

When we take the absolute value of anything, we will always end with a positive number. So, to clear the absolute value bars, we can split this into two seperate equations. Rather than

we can set two equations of

or

Our first equation, is fairly straightforward so in this equation .

Our second equation is simple to understand once we factor the minus sign. So

becomes

So add 2 to both sides. We get

Multiply both sides by , and we see that

So, since the absolute value sign means both our equations are true,

Absolute value tells you how far away a number is from 0 on the number line, so you will have to find both negative and positive values to solve this absolute value equation. To do so, you must set up two different equations.

The first one will be the positive absolute value:

.

The second one will be the "negative" absolute value. You simply add a negative sign to the left side of the equation:

Then, you solve each equation separately, leaving you with two possible answers for the value of .

or

can be rewritten as the compound statement:

or

Solve each separately to obtain the solution set:

So either or

can be rewritten as the compound statement:

or

Solve each separately to get the solution set:

So either or

First, isolate the absolute value expression on one side.

Rewrite this as the compound statement:

or

Solve each equation separately:

The equation involves an absolute value. First, we need to rewrite the equation with no absolute value.

We can split this equation into two possible equations.

Equation 1:

Equation 2:

With two equations, there are two values for . Let's start with Equation 1.

Subtract from both sides.

That's the first value for . To get the second value for , we need to repeat the steps, but with Equation 2.

Absolute value is a function that turns whatever is inside of it positive. This means that what's inside the function, , might be 7, or it could have also been -7. We have to solve for both situations.

a. subtract 1 from both sides

divide both sides by 2

b. subtract 1 from both sides

divide both sides by 2

Sometimes when you do enough absolute value problems with variables inside of them, you forget how to do ones without a variable inside. In this case, you are only going to end up with one answer because you can immediately simplify and eliminate the absolute value.

No absolute value after simplifying means you are only going to have one answer, .

First, ignore absolute value signs and solve for x.

Now, for the other solution, ignore absolute value signs and change the sign of the right side of the equation. Solve for x.

First, make sure to have the absolute value part of the equation by itself on the left side.

Now solve for both x values.

Change the side of the right hand side to find the other value of x.

The question is asking for the value of . To isolate , we must first isolate the absolute value expression.

The absolute value bars give a positive value to the result of the expression within. This means the real value of the inner expression could be positive or negative. To find the two values of , create the two possible equations.

Balance the equations to find .

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Therefore, and

Also written as .