Unit 3 ~ Contents
Algebra Beauty and Awe ~ Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3.1 Solving More Complex Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.2 Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Factors and Factoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Adding Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Quiz 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
How Wide Is a Piece of Paper? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.6 Range and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.7 Greatest Common Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.8 Complex Relationships Between Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.9 Solving Equations With Fractions or Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.10 Quiz 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
What’s the Temperature? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.11 Subtracting Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.12 Factoring the Greatest Common Factor From a Polynomial . . . . . . . . . . . . . . . . . . 35
3.13 Multiplying Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.14 Literal Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.15 Review for Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.16 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A Measure for Pleasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
, 3.1 Solving More Complex Equations
So far, you have solved equations with variables on only one side of the equation. However,
equations often have variables on both sides, such as the equation 2x + 6 = x – 1.
To solve equations with variables on both sides, simplify both sides of the equation first and then
eliminate the variable from one side of the equation using the addition or subtraction property of
equality. Then, with the variable on only one side, solve the equation using standard methods.
Example 1. Solve the equation 4x – 16 = x – 7.
4x – 16 = x – 7 Original equation.
4x – x – 16 = x – x – 7 x subtracted from both sides to eliminate the
variable from the right side.
3x – 16 = –7 Simplified.
3x – 16 + 16 = –7 + 16 16 added to both sides to eliminate the constant
from the variable side.
3x = 9 Simplified.
3x 9
= Both sides divided by 3 to change the variable
3 3
coefficient to 1.
x=3 Simplified.
Check:
4(3) – 16 = 3 – 7 3 substituted for x in the original equation.
–4 = –4 Simplified. Because this statement is true, the
answer to the equation is correct.
Simplifying an equation may result in a negative variable. Multiplying or dividing each
side by –1 will remove the negative from the variable.
It does not matter from which side the variable is eliminated. However, eliminating the
smaller of two variables gives a positive variable, which is usually preferred.
2 ~ Algebra I Unit 3
Algebra Beauty and Awe ~ Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3.1 Solving More Complex Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.2 Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Factors and Factoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Adding Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Quiz 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
How Wide Is a Piece of Paper? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.6 Range and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.7 Greatest Common Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.8 Complex Relationships Between Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.9 Solving Equations With Fractions or Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.10 Quiz 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
What’s the Temperature? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.11 Subtracting Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.12 Factoring the Greatest Common Factor From a Polynomial . . . . . . . . . . . . . . . . . . 35
3.13 Multiplying Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.14 Literal Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.15 Review for Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.16 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A Measure for Pleasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
, 3.1 Solving More Complex Equations
So far, you have solved equations with variables on only one side of the equation. However,
equations often have variables on both sides, such as the equation 2x + 6 = x – 1.
To solve equations with variables on both sides, simplify both sides of the equation first and then
eliminate the variable from one side of the equation using the addition or subtraction property of
equality. Then, with the variable on only one side, solve the equation using standard methods.
Example 1. Solve the equation 4x – 16 = x – 7.
4x – 16 = x – 7 Original equation.
4x – x – 16 = x – x – 7 x subtracted from both sides to eliminate the
variable from the right side.
3x – 16 = –7 Simplified.
3x – 16 + 16 = –7 + 16 16 added to both sides to eliminate the constant
from the variable side.
3x = 9 Simplified.
3x 9
= Both sides divided by 3 to change the variable
3 3
coefficient to 1.
x=3 Simplified.
Check:
4(3) – 16 = 3 – 7 3 substituted for x in the original equation.
–4 = –4 Simplified. Because this statement is true, the
answer to the equation is correct.
Simplifying an equation may result in a negative variable. Multiplying or dividing each
side by –1 will remove the negative from the variable.
It does not matter from which side the variable is eliminated. However, eliminating the
smaller of two variables gives a positive variable, which is usually preferred.
2 ~ Algebra I Unit 3
