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Algebra Expressions with Fraction Bars

Evaluate the numerator, evaluate the denominator, then simplify

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Algebra Expressions with Fraction Bars

Expressions Involving Fraction Bars 

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.

 

Evaluating Expressions 

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

We substitute the value for \begin{align*}z\end{align*} into the expression.

\begin{align*} \frac { 2 + 3 } { 4 } - 1\end{align*}

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:

\begin{align*} \frac { (2 + 3) }{ 4 } - 1\end{align*}

Using PEMDAS, we first evaluate the numerator, 2 + 3 = 5. Now we have:

\begin{align*}\frac { 5 } { 4 } - 1\end{align*}

We can convert \begin{align*} \frac { 5 } { 4 }\end{align*} to a mixed number:

\begin{align*} \frac { 5 } { 4 } = 1 \frac{1}{4}\end{align*}

Then evaluate the expression:

\begin{align*}\frac{5}{4}-1=1\frac{1}{4}-1=\frac{1}{4}\end{align*}

Evaluating Multi-Variable Expressions

Use the order of operations to evaluate the expression \begin{align*}\left ( \frac{ a + 2 } { b + 4 } - 1 \right ) + b\end{align*} when \begin{align*}a = 3\end{align*} and \begin{align*}b = 1.\end{align*}

Substitute the given values for \begin{align*}a\end{align*} and \begin{align*}b\end{align*} into the expression:

\begin{align*} \left ( \frac { 3 + 2 } { 1 + 4 } - 1 \right ) + 1\end{align*}

This expression effectively has nested parentheses (remember the effect of the fraction bar). The innermost grouping symbol is provided by the fraction bar. Evaluate the numerator \begin{align*}(3 + 2) = 5\end{align*} and denominator \begin{align*}(1 + 4) = 5\end{align*} first:

\begin{align*}\left ( \frac { 3 + 2 } { 1 + 4 } +1 \right ) + 1 \\ = \left ( \frac { 5 } { 5 } - 1 \right ) + 1 \qquad \text{Now evaluate inside the parentheses. First, divide.}\\ = (1 - 1) + 1 \qquad \quad \text{Next, subtract.}\\ = 0 + 1 = 1\end{align*}

Using a Graphing Calculator 

Use a graphing calculator to evaluate the expression \begin{align*} \frac { 3x^2 - 4y^2 + x^4 } { (x + y)^\frac { 1 } { 2 }}\end{align*} for \begin{align*}x = 2, \ y = -1.\end{align*}

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

Press [ENTER] to obtain the answer, \begin{align*}24\end{align*}.

 

Example

Example 1

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

Substitute the values for \begin{align*}w, \ x, \ y,\end{align*} and \begin{align*}z\end{align*} into the expression:

\begin{align*} 2 \times \left ( \frac { 11 + (3 - 2(-2)) } { (1 + 2)^2 } - 1 \right )\end{align*}

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 inside, we leave the innermost grouping symbols as parentheses \begin{align*}\left( \ \right)\end{align*}. Next will be the “invisible brackets” from the fraction bar, write these as \begin{align*}\left[ \ \right]\end{align*}. The third level of nested parentheses will be the braces \begin{align*}\left\{ \ \right\}\end{align*}. Leave negative numbers in regular parentheses.

\begin{align} & 2 \times \left \{ \frac {\left[ 11 + \left( 3 - 2 \left(-2 \right) \right) \right] } { \left [ \left( 1 + 2 \right) ^2 \right ] } - 1 \right \} \qquad \ \ \ \text{Start by evaluating the parentheses.} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \left (3 - 2(-2)\right ) = 3 + 4 = 7 \ : \ \left (1 + 2\right ) = 3\\ & = 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{Now evaluate the braces. Start with division.}\\ & = 2 \{2 - 1 \} \qquad \qquad \qquad \qquad \qquad \text{Finally, do the addition and subtraction.}\\ & = 2 \{ 1\} = 2\\ \end{align}

Review

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

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

For 4-9, evaluate the expressions by substituting for the variables.

  1. \begin{align*}\frac { jk } { j + k }\end{align*} when \begin{align*}j = 6\end{align*} and \begin{align*}k = 12\end{align*}
  2. \begin{align*}\frac{ x + y^2 } { y - x }\end{align*} when \begin{align*}x = 2\end{align*} and \begin{align*}y = 3\end{align*}
  1. \begin{align*}\frac{ 4x } { 9x^2 - 3x + 1}\end{align*} when \begin{align*}x = 2\end{align*}
  2. \begin{align*}\frac { z^2 } { x + y } + \frac{ x^2 } { x - y }\end{align*} when \begin{align*}x = 1, \ y = -2,\end{align*} and \begin{align*}z = 4\end{align*}
  3. \begin{align*}\frac { 4xyz } { y^2 - x^2 }\end{align*} when \begin{align*}x = 3, \ y = 2,\end{align*} and \begin{align*}z = 5\end{align*}
  4. \begin{align*}\frac { x^2 - z^2 } { xz - 2x(z - x)}\end{align*} when \begin{align*}x = -1 \end{align*} and \begin{align*}z = 3\end{align*}

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

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

Review (Answers)

To view the Review answers, open this PDF file and look for section 1.5. 

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Vocabulary

Fraction Bar

A fraction bar is a line used to divide the numerator and the denominator of a fraction. The fraction bar means division.

Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.

PEMDAS

PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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