2 Chapter Chapter 2 Equations, Inequalities and Problem Solving

The Addition & Multiplication Properties of Equality Section 2.2 The Addition & Multiplication Properties of Equality

Defining Linear Equations and Using the Addition Property Objective 1 Defining Linear Equations and Using the Addition Property

Linear Equations An equation is of the form “expression = expression.” An equation contains an equal sign and an expression does not. Equations Expressions Objective A

Linear Equations Linear Equation in One Variable A linear equation in one variable can be written in the form ax + b = c where a, b, and c are real numbers and a ≠ 0.

Using the Addition Property to Solve Equations Addition Property of Equality Let a, b, and c represent numbers. Then a = b and a + c = b + c are equivalent equations. Objective A

Example Solve x – 4 = 7 for x. Check: Objective A

Example Solve: y – 1.2 = –3.2 – 6.6 Objective A

Example Solve: Objective A

Example Solve: 6x + 8 – 5x = 8 – 3 x + 8 – 8 = 5 – 8 Objective A

Example Solve: 3(3x – 5) = 10x Objective A

Example Solve: 4p – 11 – p = 2 + 2p – 20 3p – 11 = 2p – 18 Simplify both sides. 3p + (– 2p) – 11 = 2p + (– 2p) – 18 Add –2p to both sides. p – 11 = –18 Simplify both sides. p – 11 + 11 = –18 + 11 Add 11 to both sides. p = –7 Simplify both sides.

Example Solve: 5(3 + z) – (8z + 9) = – 4z 15 + 5z – 8z – 9 = – 4z Use distributive property. 6 – 3z = – 4z Simplify left side. 6 – 3z + 4z = – 4z + 4z Add 4z to both sides. 6 + z = 0 Simplify both sides. 6 + (–6) + z = 0 + (–6) Add –6 to both sides. z = –6 Simplify both sides.

Using the Multiplication Property Objective Using the Multiplication Property

Example Solve: Objective A

Example Solve –4x = 16 for x. Check: Objective A

Example Solve: –1.2x = –36 Objective A

Example Solve: Multiply both sides by 7. Simplify both sides.

Using Both the Addition and Multiplication Properties Objective 3 Using Both the Addition and Multiplication Properties

Example Solve: 4x – 8x = 16 Objective A

Example Solve: 3z – 1 = 26 3z – 1 = 26 3z – 1 + 1 = 26 + 1 3z = 27

Example Solve: 12x + 30 + 8x – 6 = 10 20x + 24 = 10

Example Solve: 5(2x + 3) = –1 + 7 5(2x) + 5(3) = –1 + 7 10x + 15 = 6

Writing Word Phrases as Algebraic Expressions Objective 4 Writing Word Phrases as Algebraic Expressions

Example If x is the first two consecutive integers, express the sum of the two integers in terms of x. Simplify if possible. The second consecutive integer is always 1 more than the first. If x is the first of two consecutive integers, the two consecutive integers are x and x + 1. x + (x + 1) = 2x + 1

Example (cont) If x is the first two consecutive odd integers, express the sum of the two integers in terms of x. Simplify if possible. The second consecutive odd integer is always 2 more than the first. If x is the first of two consecutive odd integers, the two consecutive integers are x and x + 2. x + (x + 2) = 2x + 2