College Algebra Chapter 1 Equations and Inequalities Section 1.1 Linear Equations and Rational Equations Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Solve Linear Equations in One Variable Identify Conditional Equations, Identities, and Contradictions Solve Rational Equations Solve Literal Equations for a Specified Variable

Concept 1 Solve Linear Equations in One Variable

Solve Linear Equations in One Variable Step 1 Simplify both sides of the equation. Use the distributive property to clear parentheses. Combine like terms. Consider clearing fractions or decimals by multiplying both sides of the equation by the least common denominator (LCD) of all terms. Step 2 Use the addition property of equality to collect the variable terms on one side of the equation and the constant terms on the other side. Step 3 Use the multiplication property of equality to make the coefficient on the variable term equal to 1. Step 4 Check the potential solution in the original equation. Step 5 Write the solution set.

Example 1 Solve: x + 4(x – 5) = 4(x – 2)

Example 2 Solve: 2x – 3(2x + 2) = 1 – 5(4x + 3)

Skill Practice 1 Solve: 5(v – 4) – 2 = 2(v + 7) - 3

Example 3 Solve: 0.6x – 1.3 = 0.2(5x – 8)

Example 4 Solve:

Skill Practice 2 Solve:

Skill Practice 3 To rent a storage unit, a customer must pay a fixed deposit of $150 plus $52.50 in rent each month Write a model for the cost C (in $) to rent the unit for t months. If Winston has $1200 budgeted for storage , for how many months can be rent the unit?

Concept 2 Identify Conditional Equations, Identities, and Contradictions

Identify Conditional Equations, Identities, and Contradictions A conditional equation is true for some values of the variable and false for other values. An identity is an equation that is true for all values of the variable for which the expressions in an equation are defined. A contradiction is an equation that is false for all values of the variable.

Example 5 Solve: 2[4 + 2(5 – x) – 2x] = 4(7 – 2x)

Example 6 Solve: 2[4 + 2(5 – x) – 2x] = 4(7 + 2x)

Example 7 Solve: -3(x – 2) + 4(2x + 5) = 10x – (5x – 23)

Skill Practice 4 Identify each equation as a conditional equation, a contradiction, or identity. Then give the solution set. 4x+1-x=6x - 2 2(-5x-1)=2x-12x+6 2(3x-1)=6(x+1) - 8

Concept 3 Solve Rational Equations

Solve Rational Equations A rational equation is an equation in which each term contains a rational expression. All linear equations are rational equations, but not all rational equations are linear. When a variable appears in the denominator of a fraction, we must restrict the values of the variable to avoid division by zero.

Example 8 Solve: restricted value for x :________

Skill Practice 5 Solve the equation and check the solution.

Example 9 Solve: restricted value for x :________

Skill Practice 6 Solve the equation and check the solution.

Example 10 Solve: restricted values for y:________

Skill Practice 7 Solve the equation and check the solution.

Concept 4 Solve Literal Equations for a Specified Variable

Example 11 10x – 2a = 3a Solve for x.

Example 12 3x – 5y = 10 Solve for y.

Example 13 Solve for z.

Example 14 3x + 4a = 2 px +1 Solve for x.

Skill Practice 8 Solve for the indicated variable. I = Prt for t 4x + 3y = 12 for y

Skill Practice 9 Solve the equation for x. 3x – w = ax + z

Example 15 Adele's 2012 Mini-Cooper gets 37 miles per gallon on the highway and 29 miles per gallon in the city. The amount of gas she uses A (in gallons) is given by where c is the number of city miles driven and h is the number of highway miles driven. If Adele drove 58 miles in the city and used 5 gallons of gas, how many highway miles did she drive? If Adele drove 58 miles in the city and used 5 gallons of gas, how many highway miles did she drive?