Algebra - Mixed Practice (Up to Rational Expressions)

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Algebra - Mixed Practice (Up to Rational Expressions)

This is a fairly lengthy unit in most algebra textbooks. In these worksheets we'll review much of the content up to this point in the curriculum, and also cover the following topics: reducing algebraic fractions; multiplying and dividing algebraic fractions; finding least common multiples of algebraic expressions; adding and subtracting algebraic fractions; complex fractions; solving fractional equations.

Some algebra one curricula do not include fractional equations that result in quadratics. However, since we have already completed the quadratic unit, we will include some quadratic fractional equations here. Those will be notated below, so if you don't want to have your students do those problems (or if you want them to be "bonus"), you may do so.

Worksheet 7.1

Finding LCMs
Converting phrases to algebraic expressions
Reducing algebraic fractions

Worksheet 7.2

Combining like terms
Multiplying and dividing numeric fractions
Multiplying and dividing fractions

Worksheet 7.3

Solving linear equations
Factoring polynomials
Finding the LCM of algebraic expressions

Worksheet 7.4

Solving quadratic equations
Adding and subtracting numeric fractions
Adding and subtracting algebraic fractions

Worksheet 7.5

Solving word problems
Complex fractions

Worksheet 7.6

Factoring
Simplifying and factoring
Solving fractional equations (the last three problems are quadratics)

Worksheet Sets in this Series

Worksheets for this section are below the index. Click the "Overview" link to get a more detailed view of the entire series.

Lesson by Mr. Twitchell

Handouts/Worksheets

Mixed Practice 7.1

Find the LCM of 32 and 24.
Find the LCM of 20, 28, and 35.
Find the LCM of 9, 28, and 70.
Find the LCM of 2, 3, 4, 5, and 6.
Find the LCM of 7, 8, 9, and 10.

Write and simplify these algebraic expressions
Two more than three times a number
Five times seven less than a number
The sum of two more than a number and three less than a number
The product of a number and two less than the number
Five less than a number, multiplied by seven more than twice the number

Reduce the following fractions
12xy³
9xy⁴
x + 1
x² - 1
2x - 4
8
x² + 5x + 6
x + 2
3x - 9
7x - 21
x² - 4
x² + 4x + 4
x² + 7x + 12
x² + 5x + 4
9x² + 18x + 9
4x² - 4

Mixed Practice 7.1: Answer Key

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Mixed Practice 7.2

3x + 2 - 4x +5 =
(5x + 2) - (2x - 3) =
(x² - 3x + 2) - (x² + 4x - 11) =
-4(2x - 1) - 3(2 - 5x + x²) =
1
3
(6x - 3) + x(x + 4) =
2
3
·
9
12
=
1
7
·
14
3
=
9
2
÷
3
4
=
x
12
÷
3
4
=
3x
4
·
2x
5
=
x² - 9
x + 1
·
x² - 1
x - 3
2x + 12
x - 6
·
5x - 30
4x + 24
4x² + 4x + 1
x² + 3x + 2
·
x² - 1
4x² - 1
5x - 1
12
÷
25x² - 1
6
1
10x
÷
3
15x
x² + 2x + 1
x² - 4x + 4
÷
x + 1
x - 2
2 ÷
3
5
x
y²
·
x + 1
y - 1
÷
xy
y - 1

Mixed Practice 7.2: Answer Key

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Mixed Practice 7.3

Solve the equations

3x - 12x = x + 15
5(x - 12) = 3(x + 4) + 10
x(x + 1) = (x - 1)(x - 2) + 20
3x + 10 - 2(x + 1) = 15
2(x - 3(x - 1)) + 2 = 32 - 2x

Factor the following expressions
x² - 18x + 17
3x² - 12
4x² - 4xy + y²
16x⁴ - 81

Find the least common multiple of the expressions
14x³y; 21xy⁴z
x; 4x; 6x²
2x; 6x - 6
2x + 1; 2x² - x - 1
x² - 9; x² + 6x + 9
x² + 3x + 2; x² + 4x + 3
3x + 6; 4x + 8
2; x + 2; 2x - 4
9x² - 16; 3x² + 2x - 8
x² + 8x + 16; x² + 7x + 12
x³ - 4x² + 3x; 5y

Mixed Practice 7.3: Answer Key

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Mixed Practice 7.4

Solve the following equations

x² - 13x + 30 = 0
2x² - 7x = -6
2(x + 4) = x(x + 1) - 22
10x² = 21x + 27
(x + 2)(x + 3) = (x - 1)(x + 1) + 32

Simplify the fractional expressions
1
2
+
7
3
=
3 +
1
3
+
1
4
=
9
2
+ 2 =
5
6
-
1
4
=
5 -
11
3
=
x
3
+
3x
2
=
1
x
+
1
y
=
x + 1
4
+
x - 1
3
=
1
x + 1
+
2
x² + 2x + 1
=
x + 1
x² - 1
-
1
x
1
2x + 3
-
2
4x + 6
1 -
2
x + 1
1
x
+
1
x²
-
1
2x

Mixed Practice 7.4: Answer Key

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Mixed Practice 7.5

An item's price is increased by $10, and then the new price is increased by 25%. The new price is $45. What was the original price?
An item's price is increased by 25%, and then $10 is added to the price. The new price is $45. What was the original price?
The sum of three consecutive odd integers is 165. What is the largest of the numbers?
The product of two consecutive positive even integers is 168. What is the smaller of the two numbers?
Bob can drive 20 miles in a certain number of minutes. He drove 30 more miles (at the same speed) in 28 minutes. How long did his total trip take?
The number of feet in a measurement is 160 more than the number of yards in the same measurement. How many feet long is the object being measured?
Two numbers add to 15. The smaller number is 3 less than the larger. What are the two numbers?

Simplify the following fractional expressions
abc
def
bc
ef
a
b

b
c
1 +
1
2
1 -
1
2
x +
1
x
x -
1
x
a + b
c
a - b
c
1 +
2
x
+
1
x²
1 -
1
x²
x² - 9
x²
x² + 4x + 3
x