Absolutely Colorful - Absolute Value

Lesson Plan - Get It!

Audio:

Absolute value refers to the distance a number is from 0 (zero) on a number line.

One important point to remember is that an absolute value CANNOT be negative. The symbol used to represent the concept of absolute value looks like two parallel lines with a number in-between them:

|10|

When completing a math problem with absolute value, it does not matter which side of the number line the number is on. The answer to an absolute value problem is always positive.

• For example, what is |5|?
• What is |-5|?

The questions read as, "What is the absolute value of 5?" and "What is the absolute value of -5?") The best way to answer each of these questions is to draw a number line and see how far away from 0 each number is.

• What is |5|?

As you can see, the number 5 is five spaces away from 0, so the |5| = 5.

• What is |-5|?

As you can see, -5 is five spaces away from 0, so the |-5| = 5.

• What happens if you have to answer one of the following problems?

|3 + 7| or |3 - 7|

1. Step One is to simplify the problem inside of the absolute value brackets:

   • |3 + 7| = |10|

   • |3 - 7| = |-4|

2. The next step is to take the absolute value of the answer in the brackets:

   • |10| = 10

   • |-4| = 4

It's that simple! Practice with the following problems:

Now, try some more challenging problems!

We know that |-5| is 5.

• However, what is -|-5|?

Don't panic! Remember, when a number is in-between the absolute value brackets, no matter if it is a positive number or a negative number, the answer is positive. So |-5| is 5.

Now you can deal with the negative sign outside of the absolute value brackets. That negative sign changes the answer from 5 to -5.

Here are two more examples to illustrate this concept:

-|10| = -10

(The absolute value of 10 is 10. The negative sign outside of the absolute value brackets goes along with the answer to make it -10.)

-|-13|= -13

(The absolute value of -13 is 13. The negative sign outside of the absolute value brackets goes along with the answer to make it -13.)

Solve the next set of problems to practice.

Try a more challenging problem: |x| = 4

To solve this problem, rewrite it in words instead of numbers: "The absolute value of some number (x) is four."

What this equation is telling you is that the solution to this equation is four spaces away from 0.

• What number is four spaces away from zero?

If you answered "4," then you are right. Great job!

However, that is not the only answer to this equation. There is another number that is 4 spaces away from 0 on a number line.

• Do you know what it is?

If you answered "-4," then tell yourself that you are cool because you are, and your answer is correct!

Continue to challenge yourself with the following problem:

|x + 6|= 7

Before you try to solve this equation, take a moment to put the equation into words so you can fully understand how to solve this equation.

The problem, in words, would be, "The absolute value of some number plus six is seven." If you break it down even more, you could say, "The answer to the equation in the absolute value brackets is 7 spaces away from zero."

• So, what numbers are seven spaces away from zero on the number line?

Right, 7 and -7. That is not the solution to the equation. It is just a step in the right direction.

Follow the steps below to completely solve the equation |x + 6|= 7:

1. Step One: Rewrite your problem without the absolute value brackets:

x + 6 = 7

• Does this equation look familiar?

When learning how to solve one-step equations, the problems were written the same way.

2. Step Two: Now that you have dropped the absolute value brackets, you need to write the problem two ways.

The first way to re-write the equation is x + 6 = 7. This is the original problem without the absolute value brackets.

The second way to re-write the equation is x + 6 = -7.

The reason you have to write the equation two different ways is that you need to solve the equations for both 7 and -7. Remember, whatever is inside the absolute value brackets can have a value that is positive or negative. So in this case, x + 6 can equal 7 or -7, since both numbers are an equal distance from 0.

Now, it is time to solve both of the equations.

3. Step Three: Solve x + 6 = 7

Remember, when solving equations, you need to get the variable alone on one side of the equal sign and the constant(s) on the other side.

For this equation, the way to complete this is to subtract 6 from both sides.

• x + 6 - 6 = 7 - 6
• x = 1

After completing the operations, you are left with the solution x = 1.

4. Step Four: Now, solve the second equation, x + 6 = -7

• x + 6 - 6 = -7 - 6
• x = -13

The solution to the second equation is x = -13.

5. Step Five: Go back to each of your equations and replace the x with your answers. Calculate each equation to see if your solution is correct.

• x + 6 = 7
• 1 + 6 = 7
• 7 = 7

Your first solution is correct!

• x + 6 = -7
• -13 + 6 = -7
• -7 = -7

Your second solution is correct!

6. Step Six: Put your answers next to your original equation:

|x + 6| = 7 (1, -13) Put the equation with the solutions in it into words to help you better understand why the two solutions make sense.

"The absolute value of one plus 6 is 7." "The absolute value of negative 13 plus 6 is 7."

Just a reminder, it does not matter if the number inside of the absolute value brackets is positive or negative, the absolute value of that number is always positive.

Try a few problems below for practice!

Here is a video called, Absolute value and number lines | Negative numbers and absolute value | Pre-Algebra | Kahn Academy. Take time to watch it to review the skills you just learned:

When you are ready, continue on to the Got It? section for some more practice.