University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

437

Pages:

12
Author:

vergue7oj

Chapter 9 Equations, Inequalities and Problem Solving Part 4 Chapter Sections 9.1 – The Addition Property of Equality 9.2 – The Multiplication Property of Equality 9.3 – Further Solving Linear Equations 9.4 – Introduction to Problem Solving 9.5 – Formulas and Problem Solving 9.6 – Percent and Mixture Problem Solving 9.7 – Solving Linear Inequalities 2 § 9.7 Solving Linear Inequalities Linear Inequalities A linear inequality in one variable is an equation that can be written in the form ax + b < c • a, b, and c are real numbers, a ≠ 0 • < symbol could be replaced by > or ≤ or ≥ 51 Graphing Solutions Graphing solutions to linear inequalities in one variable (using circles) • Use a number line • Use a closed circle at the endpoint of a interval if you want to include the point • Use an open circle at the endpoint if you DO NOT want to include the point 7 -4 Represents the set {xx ≤ 7} Represents the set {xx > – 4} 52 Linear Inequalities Graphing solutions to linear inequalities in one variable (using interval notation) • Use a number line • Use a bracket at the endpoint of a interval if you want to include the point • Use a parenthesis at the endpoint if you DO NOT want to include the point Interval Notation ] ( -4 7 Represents the set (– ∞, 7] Represents the set (– 4, ∞) 53 Properties of Inequality Addition Property of Inequality • a < b and a + c < b + c are equivalent inequalities Multiplication Property of Inequality • a < b and ac < bc are equivalent inequalities, if c is positive • a < b and ac > bc are equivalent inequalities, if c is negative 54 Solving Linear Inequalities Solving linear inequalities in one variable 1) 2) 3) 4) Multiply to clear fractions Use distributive property Simplify each side of equation Get all variable terms on one side and numbers on the other side of equation (addition property of equality) 5) Isolate variable (multiplication property of equality) 55 Solving Linear Inequalities Example 3x + 9 ≥ 5(x – 1) 3x + 9 ≥ 5x – 5 3x – 3x + 9 ≥ 5x – 3x – 5 (Use distributive property on right side) (Subtract 3x from both sides) 9 ≥ 2x – 5 (Simplify both sides) 9 + 5 ≥ 2x – 5 + 5 (Add 5 to both sides) 14 ≥ 2x 7≥x Graph of solution (– ∞,7] (Simplify both sides) (Divide both sides by 2) ] 7 56 Solving Linear Inequalities Example 7(x – 2) + x > – 4(5 – x) – 12 7x – 14 + x > – 20 + 4x – 12 8x – 14 > 4x – 32 8x – 4x – 14 > 4x – 4x – 32 4x – 14 > –32 4x – 14 + 14 > –32 + 14 4x > –18 x> −9 2 Graph of solution ( -9 ,∞) 2 (Use distributive property) (Simplify both sides) (Subtract 4x from both sides) (Simplify both sides) (Add 14 to both sides) (Simplify both sides) (Divide both sides by 4 and simplify) ( -9 2 57 Compound Inequalities A compound inequality is two inequalities joined together. 0 ≤ 4(5 – x) < 8 To solve the compound inequality, perform operations simultaneously to all three parts of the inequality (left, middle and right). 58 Solving Compound Inequalities Example 0 ≤ 4(5 – x) < 8 0 ≤ 20 – 4x < 8 0 – 20 ≤ 20 – 20 – 4x < 8 – 20 (Use the distributive property) (Subtract 20 from each part) – 20 ≤ – 4x < – 12 (Simplify each part) 5≥x>3 (Divide each part by –4) Remember that the sign direction changes when you divide by a number < 0! Graph of solution (3,5] ( 3 ] 5 59