Solving Absolute Value Equations - Algebra 2

Solve the following equation for b:

Let's begin by subtracting 6 from both sides:

Next, we can get rid of the absolute value signs and make our two equations:

Next, add 13 to both sides

And divide by 5:

or

Next, add 13 to both sides

And divide by 5:

So our answers are

Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.

To solve this you need to set up two different equations then solve for x.

The first one is:

where

The other equations is:

where

Recall the general definition of absolute value,

We will attempt two cases for the absolute value,

Case 1:

Start by isolating the abolute value term,

Replace with and solve for :

Case 2:

Replace with and solve for :

Testing the Solutions

Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.

For instance, if we test in the original equation we get,

This is clearly not true, since both sides are not equal, this rules out as a solution. Similiarly, using you can show that it also fails. Therefore, there is no solution.

If , then or

Solve each of those equations to find the possible values for x.

or

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract on both sides.

Distribute the negative sign to each term in the parenthesEs.

Add on both sides.

Divide both sides by .

Answers are

When dealing with absolute value equations, we need to deal with negative values as well.

Add on both sides.

Subtract on both sides.

Distribute the negative sign to each term in the parentheses.

Add on both sides.

Divide on both sides.

Answers are

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract on both sides.

Divide on both sides.

To solve an absolute value equation, one must first rewrite the problem two different ways.

One is

and the other is

.

Then, solve each equation for v.

Your answers are 8 and -5.

To solve absolute value equations, first we must isolate the absolute value:

Now that the absolute value is the only thing on the left hand side of the equation, we can solve for x.

Keep in mind that the absolute value makes whatever is inside of it a positive value. This means that and are both valid in solving the equation.

Setting both of these equal to 3 and solving for x, we get

As a check, plug each of these solutions back into the original equation and see if both solutions are valid!

The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, can never happen. There is no solution.

Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.

First, subtract 5 from both sides to get the absolute value expression alone.

Split this into two linear equations:

or

The solution set is

Divide both sides by 3.

Consider both the negative and positive values for the absolute value term.

Subtract 2 from both sides to solve both scenarios for .

The absolute value gives two problems to solve. Remember to switch the "less than" to "greater than" when comparing the negative term.

or

Solve each inequality separately by adding to all sides.

or

This can be simplified to the format .

Remove the absolute value by setting the term equal to either or . Remember to flip the inequality for the negative term!

Solve each scenario independently by subtracting from both sides.

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

and

This gives us:

and

However, this question has an outside of the absolute value expression, in this case . Thus, any negative value of will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus, is an extraneous solution, as cannot equal a negative number.

Our final solution is then

The absolute value of any number is nonnegative, so must always be greater than . Therefore, any value of makes this a true statement.