2 Equations, Inequalities, and Applications

2.1 The Addition Property of Equality Objectives 1. Identify linear equations. 2. Use the addition property of equality. 3. Simplify, and then use the addition property of equality.

Identify 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, with A ≠ 0. Some examples of linear and nonlinear equations follow. 4x + 9 = 0, 2x – 3 = 5, and x = 7 Linear x2 + 2x = 5, = 6, and |2x + 6| = 0 Nonlinear

Identify Linear Equations A solution of an equation is a number that makes the equation true when it replaces the variable. Equations that have exactly the same solution sets are equivalent equations. A linear equation is solved by using a series of steps to produce a simpler equivalent equation of the form x = a number or a number = x.

Use the Addition Property of Equality If A, B, and C are real numbers, then the equations A = B and A + C = B + C are equivalent equations. In words, we can add the same number to each side of an equation without changing the solution set.

Use the Addition Property of Equality Note Equations can be thought of in terms of a balance. Thus, adding the same quantity to each side does not affect the balance.

Use the Addition Property of Equality Example 1 Solve the equation. Our goal is to get an equivalent equation of the form x = a number. x – 23 = 8 x – 23 + 23 = 8 + 23 x = 31 Check: 31 – 23 = 8 

Use the Addition Property of Equality Example 2 Solve the equation. y – 2.7 = –4.1 y – 2.7 + 2.7 = –4.1 + 2.7 y = – 1.4 Check: –1.4 – 2.7 = –4.1 

Use the Addition Property of Equality The same number may be subtracted from each side of an equation without changing the solution. If a is a number and –x = a, then x = –a.

Use the Addition Property of Equality Example 3 Solve the equation. Our goal is to get an equivalent equation of the form x = a number. –12 = z + 5 –12 – 5 = z + 5 – 5 –17 = z Check: –12 = –17 + 5 

Subtracting a Variable Term Example 4 Solve the equation. 4a + 8 = 3a 4a – 4a + 8 = 3a – 4a 8 = –a –8 = a Check: 4(–8) + 8 = 3(–8) ? –24 = –24 

Simplify and Use the Addition Property of Equality Example 7 Solve. Check: 5((2 · –36) –3) – (11(–36) + 1) = 20 5(–72 –3) – (–396 + 1) = 20 5(2b – 3) – (11b + 1) = 20 5(–75) – (–395) = 20 10b – 15 – 11b – 1 = 20 –375 + 395 = 20 –b – 16 = 20 20 = 20 –b – 16 + 16 = 20 + 16  –b = 36 b = –36