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# Algebra Expressions with Fraction Bars

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What if you had a mathematical expression containing fraction bars, like $\frac { (7 - 3)^2 } { 6 - 4 } + 5$ ? How could you find its value? After completing this Concept, you'll be able to use the order of operations to evaluate expressions like this one.

### Try This

For more practice, you can play an algebra game involving order of operations online at http://www.funbrain.com/algebra/index.html .

### Guidance

Fraction bars count as grouping symbols for PEMDAS , so we evaluate them in the first step of solving an expression. All numerators and all denominators can be treated as if they have invisible parentheses around them. When real parentheses are also present, remember that the innermost grouping symbols come first. If, for example, parentheses appear on a numerator, they would take precedence over the fraction bar. If the parentheses appear outside of the fraction, then the fraction bar takes precedence.

#### Example A

Use the order of operations to evaluate the following expression: $\frac { z + 3 } { 4 } - 1$ when $z = 2$

Solution:

We substitute the value for $z$ into the expression.

$\frac { 2 + 3 } { 4 } - 1$

Although this expression has no parentheses, the fraction bar is also a grouping symbol—it has the same effect as a set of parentheses. We can write in the “invisible parentheses” for clarity:

$\frac { (2 + 3) }{ 4 } - 1$

Using PEMDAS , we first evaluate the numerator:

$\frac { 5 } { 4 } - 1$

We can convert $\frac { 5 } { 4 }$ to a mixed number:

$\frac { 5 } { 4 } = 1 \frac{1}{4}$

Then evaluate the expression:

$\frac{5}{4}-1=1\frac{1}{4}-1=\frac{1}{4}$

#### Example B

Use the order of operations to evaluate the following expression: $\left ( \frac{ a + 2 } { b + 4 } - 1 \right ) + b$ when $a = 3$ and $b = 1$

Solution:

We substitute the values for $a$ and $b$ into the expression:

$\left ( \frac { 3 + 2 } { 1 + 4 } - 1 \right ) + 1$

This expression has nested parentheses (remember the effect of the fraction bar). The innermost grouping symbol is provided by the fraction bar. We evaluate the numerator $(3 + 2)$ and denominator $(1 + 4)$ first.

$\left ( \frac { 3 + 2 } { 1 + 4 } +1 \right ) + 1 & = \left ( \frac { 5 } { 5 } - 1 \right ) + 1 \qquad \text{Next we evaluate the inside of the parentheses. First we divide.}\\& = (1 - 1) + 1 \qquad \ \ \text{Next we subtract.}\\& = 0 + 1 = 1$

#### Example C

Evaluate the expression $\frac { 3x^2 - 4y^2 + x^4 } { (x + y)^\frac { 1 } { 2 }}$ for $x = 2, \ y = -1$ .

Solution

Store the values of $x$ and $y$ : 2 [STO] $x$ , -1 [STO] $y$ . (The letters $x$ and $y$ can be entered using [ALPHA] + [KEY] .) Input the expression in the calculator. When an expression includes a fraction, be sure to use parentheses: $\frac{(\text{numerator})}{(\text{denominator})}$ .

Press [ENTER] to obtain the answer $24$ .

Watch this video for help with the Examples above.

### Vocabulary

• Fraction bars count as grouping symbols for PEMDAS , so we evaluate them in the first step of solving an expression. All numerators and all denominators can be treated as if they have invisible parentheses around them.

### Guided Practice

Use the order of operations to evaluate the following expression: $2 \times \left ( \frac { w + (x - 2z) } {(y + 2)^2} - 1 \right )$ when $w = 11, x = 3, y = 1,$ and $z = -2$

Solution:

We substitute the values for $w, x, y,$ and $z$ into the expression:

$2 \times \left ( \frac { 11 + (3 - 2(-2)) } { (1 + 2)^2 } - 1 \right )$

This complicated expression has several layers of nested parentheses. One method for ensuring that we start with the innermost parentheses is to use more than one type of parentheses. Working from the outside, we can leave the outermost brackets as parentheses $( )$ . Next will be the “invisible brackets” from the fraction bar; we will write these as $[ \ ]$ . The third level of nested parentheses will be the { }. We will leave negative numbers in round brackets.

$& 2 \times \left ( \frac {[11 + \left \{3 - 2(-2)\right \}] } { \left [ \left \{1 + 2\right \}^2 \right ] } - 1 \right ) \qquad \text{Start with the innermost grouping sign:} \ \left \{ \right \}. \\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \left \{1 + 2\right \} = 3; \ \left \{3 - 2(-2)\right \} = 3 + 4 = 7\\& = 2 \left ( \frac{ [11 + 7] } { [3^2] } - 1 \right ) \qquad \qquad \qquad \text{Next, evaluate the square brackets.}\\& = 2 \left ( \frac { 18 } { 9 } - 1 \right ) \qquad \qquad \qquad \qquad \ \text{Next, evaluate the round brackets. Start with division.}\\& =2(2 - 1) \qquad \qquad \qquad \qquad \quad \ \ \text{Finally, do the addition and subtraction.}\\& =2(1) = 2$

### Practice

For 1-3, use the order of operations to evaluate the following expressions.

1. $\frac {2 \cdot (3 + (2 - 1)) } { 4 - (6 + 2) } - (3 - 5)$
2. $\frac { 4 + 7(3) } { 9 - 4 } + \frac { 12 - 3 \cdot 2 } { 2 }$
3. $\frac { (2^2 + 5)^2} { 5^2 - 4^2} \div (2 + 1)$

For 4-5, evaluate the following expressions involving variables.

1. $\frac { jk } { j + k }$ when $j = 6$ and $k = 12$
2. $\frac{ x + y^2 } { y - x }$ when $x = 2$ and $y = 3$

For 6-9, evaluate the following expressions involving variables.

1. $\frac{ 4x } { 9x^2 - 3x + 1}$ when $x = 2$
2. $\frac { z^2 } { x + y } + \frac{ x^2 } { x - y }$ when $x = 1, \ y = -2,$ and $z = 4$
3. $\frac { 4xyz } { y^2 - x^2 }$ when $x = 3, \ y = 2,$ and $z = 5$
4. $\frac { x^2 - z^2 } { xz - 2x(z - x)}$ when $x = -1$ and $z = 3$

For 10-14, evaluate each expression using a graphing calculator.

1. $x^2 + 2x - xy$ when $x = 250$ and $y = -120$
2. $(xy - y^4)^2$ when $x = 0.02$ and $y = -0.025$
3. $\frac { x + y - z } { xy + yz + xz }$ when $x = \frac { 1 } { 2 }, \ y = \frac{3}{2},$ and $z = -1$
4. $\frac{(x + y)^2}{4x^2 - y^2}$ when $x = 3$ and $y = -5$
5. $\frac{(x - y)^3}{x^3 - y} + \frac{(x + y)^2}{x + y^4}$ when $x = 4$ and $y = -2$