Solving Absolute Value Equations

The absolute value of a number is the distance from the number to zero. The absolute of 2 is just 2 because the distance between 2 and 0 is 2. The absolute value of -2 is also 2 because the distance between -2 and 0 is 2. In other words, the absolute value of a positive number is just the number and the absolute value of a negative number is just the number with the negative sign left off. For example, the
absolute value of 5 is 5, and the absolute value of -5 is 5. The letter x in algebra also has an absolute value. The absolute value of x is x, and the absolute value of -x is x. Whenever you put two bars around a number, it stands for the absolute value of the number. If you see I 3 I, it stands for the absolute value of 3, which is 3.

Let’s try to solve an absolute value equation.

I x I = 1.

This equation says that the absolute value of x is 1. We need to find out what x equals. There are two numbers that have an absolute value of 1. Both 1 and -1 have an absolute value of 1. That means that x could equal 1 or -1. Let’s plug 1 and -1 into the equation to see if they both make the equation true.

I 1 I = 1

I – 1 I = 1

Both of these equations are true. The first one says that the absolute value of 1 is 1 and the second one says that the absolute value of -1 is 1. That means we have two answers. The first answer is x = 1, and the second answer is x = -1. This means there is a strategy we can use to solve this type of problem.

I x I = 1

We can rewrite this problem with two equations.

x = 1

x = -1

This is the strategy. First, you take away the absolute value bars. Next, you set up two equations. The first one has a positive number on the right side and the second one has a negative sign next to the number on the right side. We can use this strategy to solve a more complicated problem.

I x + 9 I = 11

Let’s rewrite the problem with two separate equations.

x + 9 = 11

x + 9 = -11

Now we just solve both equations. In the first equation, we subtract 9 from both sides and we have x = 2. 9 – 9 equals 0 on the left side, and 11 – 9 equals 2 on the right side. In the second equation, we subtract 9 from both sides and we have x = -20. 9 – 9 = 0 on the left side, and -11 – 9 = -20 on the right side. Our two answers are x = 2 and x = -20. Let’s check our answers by plugging 2 and -20 into the equation.

I 2 + 9 I = 11.

This one comes out to:

I 11 I = 11

This is true because the absolute value of 11 is 11.

I -20 + 9 I = 11

This one comes out to:

I -11 I = 11

This is also true because the absolute value of -11 is 11.

Both answers are correct. The two answers are x = 2 and x = -20.


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