Have you ever had a part time job?

Lydia and Bart both work at a bookstore. Lydia makes \begin{align*}x\end{align*} amount each hour; Bart has more experience, so he makes 1.5 times more than Lydia makes each hour. If Lydia and Bart both work 4 hours a day, how much will they make together in seven days if Lydia makes $8 an hour.

Write an expression; then solve it. Do you know how to do this?

**This Concept will show you exactly how to do this.**

### Guidance

Previously we worked on the basic order of operations. Well, now we can expand our rules to include evaluation of more complicated expressions. In intricate expressions ** parentheses** are used as grouping symbols. Parentheses indicate which operations should be done first. In the order of operations, operations within the parentheses are

*always*first.

In the Evaluate Powers with Variable Bases Concept, we learned how to simplify and evaluate exponents.

\begin{align*}x^3 = xxx\end{align*}.

Exponential notation is another factor we must account for in the order of operations. After completing the operations in parentheses, we then evaluate the exponents. Then we complete the multiplication and division from left to right; finally, we complete the addition and subtraction from left to right. The chart below shows the complete order of operations. If you keep the order of operations in the forefront of your mind and are careful to take each step one at a time and show your work, evaluating complex expressions using the order of operations will become second nature.

**Let’s look at the order of operation once again.**

Evaluate the expression: \begin{align*}2x^2-(x+7)\end{align*} *if* \begin{align*}x=4\end{align*}

**First, we substitute 4 in for \begin{align*}x\end{align*}.**

\begin{align*}2(4^2)-(4+7)\end{align*}

**Next, we add the terms in parentheses. Since parentheses is the first order of operations.**

\begin{align*}2(4^2)-(11)\end{align*}

**Now we can evaluate the exponent.**

\begin{align*}2(16) - 11\end{align*}

**Multiplication is next and there is no division.**

\begin{align*}32 - 11\end{align*}

**Finally, we complete with subtraction since there is no addition.**

**21**

**Our answer is 21.**

**Sometimes you will also evaluate expressions with more than one variable. Just keep track and follow the order of operations and you will be all set.**

Now it's time for you to practice.

#### Example A

Evaluate \begin{align*}3x^2-2+(x+3)\end{align*} ** if** \begin{align*}x\end{align*}

*is***2**

**Solution: 15**

#### Example B

Evaluate \begin{align*}\frac{24}{x}+ (9-x)+y^2\end{align*} ** if** \begin{align*}x\end{align*}

*is***3**

**\begin{align*}y\end{align*}**

*and*

*is***4**

**Solution: 30**

#### Example C

Evaluate \begin{align*}5x^2-2+(3+3)\end{align*} ** if** \begin{align*}x\end{align*}

*is***5**

**Solution: 129**

Now back to the dilemma at the bookstore. Here is the original problem once again.

Lydia and Bart both work at a bookstore. Lydia makes \begin{align*}x\end{align*} amount each hour; Bart has more experience, so he makes 1.5 times more than Lydia makes each hour. If Lydia and Bart both work 4 hours a day, how much will they make together in seven days if Lydia makes $8 an hour.

Write an expression; then solve it.

In this problem, \begin{align*}x\end{align*} stands for the amount Lydia makes per hour. Because Bart makes 1.5 times the amount that Lydia makes per hour, then \begin{align*}1.5x\end{align*} describes how much Bart makes per hour. The total Lydia and Bart make in 1 hour is therefore \begin{align*}(x + 1.5x)\end{align*}. Now the problem tells us that Lydia and Bart each work 4 hours per day. So, the amount they both make in 1 day is \begin{align*}4(x + 1.5x)\end{align*}. We want to find out how much they make in 7 days, so our expression is \begin{align*}7 \times 4(x + 1.5x)\end{align*}. The problem gives us the value of \begin{align*}x\end{align*}. Lydia makes $8 an hour. We substitute 8 for the variable in our expression and solve using the order of operations. Remember: parenthesis, exponents, multiplication, division, addition, subtraction.

\begin{align*}7 \times 4(x + 1.5x) &= 7 \times 4(8 + 1.5 \times 8)\\ 7 \times 4(x + 1.5x) &= 7 \times 4(8 + 12)\\ 7 \times 4(x + 1.5x) &= 7 \times 4(20)\\ 7 \times 4(x + 1.5x) &= 560\end{align*}

**The answer is $560.00.**

### Vocabulary

- Numerical Expression
- an expression that uses numbers and operations.

- Variable Expression
- an expression that uses numbers, variables and operations.

- Parentheses
- grouping symbols, the first step of the order of operations.

- Exponent
- the little number that tells how many times to multiply the base times itself.

- Order of Operations
- the order that you perform each operation when evaluating an expression.

### Guided Practice

Here is one for you to try on your own.

Evaluate \begin{align*}6x^2-2x+(x+3)\end{align*} ** if** \begin{align*}x\end{align*}

*is***4**

**Answer**

First, substitute 4 into every place that an \begin{align*}x\end{align*} appears.

\begin{align*}6(4)^2-2(4)+(4+3)\end{align*}

Now evaluate according to the order of operations.

**The answer is 95.**

### Video Review

- This is a James Sousa video on evaluating an expressions using the order of operations.

### Practice

Directions: Evaluate the following variable expressions is \begin{align*}x=4, y=2, z=3\end{align*}

1. \begin{align*}x^2+ y\end{align*}

2. \begin{align*}2y^2+ 5-2\end{align*}

3. \begin{align*}x^2- y^2+ z\end{align*}

4. \begin{align*}3x^2+ 2x^2\end{align*}

5. \begin{align*}8+ x^2- 4y\end{align*}

6. \begin{align*}14 \div 2+ z^2- y\end{align*}

7. \begin{align*}20 + z^2-y\end{align*}

8. \begin{align*}5x-2y+3z\end{align*}

9. \begin{align*}5+(x-z)+ 5(6)\end{align*}

10. \begin{align*}8 + x-y^2+z\end{align*}

11. \begin{align*}(x+y)+ 5 \cdot 2 - 3\end{align*}

12. \begin{align*}4x^2+3z^3+ 2\end{align*}

Directions: Use what you have learned to answer the following questions true or false.

13. Parentheses are a grouping symbol.

14. Exponents can’t be evaluated unless the exponent is equal to 3.

15. If there is multiplication and division in a problem you always do the division first.