# 2.9: Inverse Property of Addition in Decimal Equations

**At Grade**Created by: CK-12

**Practice**Inverse Property of Addition in Decimal Equations

Have you ever run a half marathon? Well, Kelly ran one and then ran a 5k too. Take a look.

Kelly ran a half marathon in 2.01 hours. She ran a 5k on the next day. Her total time for both races was 2.29.

What was Kelly's time for the 5k?

Can you write an equation with a variable that shows these values?

Then can you use the inverse property of addition to solve it?

**Use this Concept to learn all about the inverse property of addition in decimal situations. Then you will be able to accomplish this task.**

### Guidance

**The word “***inverse***” is one that is used in mathematics.**

In fact, as you get into higher levels of math, you will see the word “inverse” used more often.

**What do we mean when we use the word inverse?**

**The word inverse means opposite.** Sometimes numbers are opposites of each other, and sometimes operations are opposites of each other too. Subtraction is called the ** inverse**, or opposite, of addition. If you’ve ever used addition to check subtraction calculations, you already have a certain idea of the relationship between addition and subtraction.

**The** *inverse property of addition***states that for every number \begin{align*}x, x + (-x) = 0\end{align*}.** In other words, when you add \begin{align*}x\end{align*} to its opposite, \begin{align*}-x\end{align*} (also known as the ** additive inverse**), the result is zero.

*Adding***a** *negative***number is the same as subtracting that number.**

**Let’s put whole numbers in place of \begin{align*}x\end{align*} to make the property clear: \begin{align*}3 + (-3) = 0\end{align*}.**

The inverse property of addition may seem obvious, but it has important implications for our ability to solve variable equations which aren’t easily solved using mental math—such as variable equations involving decimals.

**Remember back to our work with equations and expressions in an earlier Concept?**

Think about an equation. An equation is a number sentence where one expression is equal to another expression. The inverse property of addition lets us add or subtract the same number to both sides of the equation without changing the solution to the equation. This technique is called an ** inverse operation** and it lets us get the variable \begin{align*}x\end{align*} alone on one side of the equation so that we can find its value. Take a look at how it’s done.

\begin{align*}x + 5 & = 15\\ x + 5 - 5 & = 15 - 5 \rightarrow \text{inverse operation means we subtract} \ 5 \ \text{from both sides}\\ x & = 10\end{align*}

Notice how, on the left side of the equation, \begin{align*}+ 5 - 5 = 0\end{align*}, and cancels itself out leaving the \begin{align*}x\end{align*} alone on the left side.

**Our answer is that \begin{align*}x\end{align*} is equal to 10.**

**Now let’s look at one where the two expressions in the equation are decimals.**

\begin{align*}x + 39.517 = 50.281\end{align*}

**First, we need to isolate the variable. To do this, we can use the inverse operation. This is an addition problem, so we use subtraction to get the variable by itself. We subtract 39.517 from both sides of the equation.**

\begin{align*}x + 39.517 & = 50.281\\ x + 39.517 - 39.517 & = 50.281 - 39.517 \rightarrow \text{inverse operations: subtract} \ 39.517 \ \text{from both sides}\\ x & = 10.764\end{align*}

**Notice how we subtracted 39.517 from both sides of the equation.** Remember, both expressions on either side of an equation must be equal *at all times*. Whatever operation you perform to one side of the equation, you must perform to the other side.

**In this instance, on the left side of the equation, \begin{align*}+ 39.517 - 39.517 = 0\end{align*}, and cancels itself out leaving the \begin{align*}x\end{align*} alone on the left side. On the right side of the equation, 50.281 - 39.517 gives us the value of \begin{align*}x\end{align*}.**

**Our answer is that \begin{align*}x\end{align*} is equal to 10.764.**

\begin{align*}x - 43.27 = 182.205\end{align*}

**First, we need to isolate the variable. To do this, we can use the inverse operation. Notice that this is a subtraction problem. To isolate the variable we are going to use the inverse of subtraction which is addition.**

\begin{align*}x - 43.27 & = 182.205\\ x - 43.27 + 43.27 & = 182.205 + 43.27 \rightarrow \text{inverse operations: add} \ 43.27 \ \text{to both sides}\\ x & = 225.475\end{align*}

**Notice how we added 43.27 to both sides of the equation**. Remember, both expressions on either side of an equation must be equal *at all times*. **Whatever operation you perform to one side of the equation, you must perform to the other side.** In this instance on the left side of the equation \begin{align*}-43.27 + 43.27 = 0\end{align*}, and cancels itself out leaving the \begin{align*}x\end{align*} alone on the left side. On the right side of the equation, 182.205 + 43.27 gives us the value of \begin{align*}x\end{align*}.

**Our answer is that \begin{align*}x\end{align*} is equal to 225.475.**

Use the inverse property of addition to solve each equation.

#### Example A

\begin{align*}x+5.678=12.765\end{align*}

**Solution: \begin{align*}7.087\end{align*}**

#### Example B

\begin{align*}x-4.32=19.87\end{align*}

**Solution: \begin{align*}24.19\end{align*}**

#### Example C

\begin{align*}x+123.578=469.333\end{align*}

**Solution: \begin{align*}345.755\end{align*}**

Here is the original problem once again. Use what you have learned to solve it.

Kelly ran a half marathon in 2.01 hours. She ran a 5k on the next day. Her total time for both races was 2.29.

What was Kelly's time for the 5k?

Can you write an equation with a variable that shows these values?

Then can you use the inverse property of addition to solve it?

First, let's write an equation using the given information. We don't know the time for the 5k, so this is our variable.

\begin{align*}x + 2.01 = 2.29\end{align*}

Now we can use the inverse operation for addition to solve.

\begin{align*}x = 2.29 - 2.01\end{align*}

\begin{align*}x = .28\end{align*}

**Kelly ran the 5k in 28 minutes.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Inverse
- means opposite

- Inverse Property of Addition
- means that when you add two opposite values together that the answer is 0.

- Additive Inverse
- using the opposite operation to solve for an unknown variable in an equation.

### Guided Practice

Here is one for you to try on your own.

At Saturday’s track meet, Liz ran 1.96 kilometers less than Sonya. Liz ran 1.258 kilometers. How many kilometers did Sonya run?

**Answer**

Anytime you see the key words “less than,” you know you’re going to write a subtraction equation.

In this problem, we are trying to find out the number of kilometers Sonya ran. We’ll give that value the variable \begin{align*}x\end{align*}. We know how many kilometers Liz ran, and we know the subtraction relationship between the Liz’s distance and Sonya’s distance, so we can write the equation \begin{align*}x - 1.96 = 1.258\end{align*}. Now we can solve for the value of \begin{align*}x\end{align*} using inverse operations.

\begin{align*}x - 1.96 & = 1.258\\ x - 1.96 + 1.96 & = 1.258 + 1.96 \rightarrow \text{inverse operation, add} \ 1.96 \ \text{to both sides}\\ x & = 3.218\end{align*}

*Remember to put the units of measurement in your answer!*

**The answer is 3.218 kilometers.**

### Video Review

Here is a video for review.

- This is a Khan Academy video on the inverse property of addition.

### Practice

Directions: Solve each equation for the missing variable using the inverse operation.

1. \begin{align*}x + 2.39 = 7.01\end{align*}

2. \begin{align*}x + 5.64 = 17.22\end{align*}

3. \begin{align*}x + 8.07 = 18.l2\end{align*}

4. \begin{align*}x + 14.39 = 7.342\end{align*}

5. \begin{align*}x + 21.3 = 87.12\end{align*}

6. \begin{align*}x + 31.9 = 7.22\end{align*}

7. \begin{align*}x + 18.77 = 97.12\end{align*}

8. \begin{align*}x + 21.31 = 27.09\end{align*}

9. \begin{align*}x + 18.11 = 87.22\end{align*}

10. \begin{align*}818.703 = 614.208 + x\end{align*}

11. \begin{align*}x + 55.27 = 100.95\end{align*}

Directions: Use equations to solve each word problem. Each answer should have an equation and a value for the variable.

12. Jamal’s leek and potato soup calls for 2.45 kg more potatoes than leeks. Jamal uses 4.05 kg of potatoes. How many kilograms of leeks does he use? Write an equation and solve.

13. He distance from Waterville to Longford is 118.816 kilometers less than the distance from Transtown to Longford. The distance from Waterville to Longford is 67.729 kilometers. What is the distance from Transtown to Longford? Write an equation and solve.

14. Sabrina spent $25.62 at the book fair. When she left the fair, she had $6.87. How much did money did she take to the fair? Write an equation and solve.

15. Mr. Bodin has 11.09 liters of a cleaning solution, which is a combination of soap and water. If there are 2.75 liters of soap in the solution, how many liters of water are in the solution? Write an equation and solve.

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### Image Attributions

Here you'll learn to identify and apply the inverse property of addition in decimal operations with variable and numerical expressions.