2 Equations, Inequalities, and Applications

2.7 Solving Linear Inequalities Objectives 1. Graph intervals on a number line. 2. Use the addition property of inequality. 3. Use the multiplication property of inequality. 4. Solve inequalities using both properties of inequality. 5. Solve linear inequalities with three parts. 6. Use inequalities to solve applied problems.

Inequalities are algebraic expressions related by < “is less than,” ≤ “is less than or equal to,” > “is greater than,” ≥ “is greater than or equal to.” We solve an inequality by finding all real number solutions for it. For example, the solutions of x ≤ 2 include all real numbers that are less than or equal to 2, not just the integers less than or equal to 2.

Examples of linear inequalities in one variable include Linear Inequality in One Variable A linear inequality in one variable can be written in the form Ax + B < C, where A, B, and C are real numbers, with A ≠ 0. Examples of linear inequalities in one variable include x + 5 < 2, t – 3 ≥ 5, and 2k + 5 ≤ 10.

Graph Intervals on a Number Line Example 1 Graph x > –2. The statement x > –2 says that x can represent any value greater than –2 but cannot equal –2 itself. We show this interval on a graph by placing a parenthesis at –2 and drawing an arrow to the right. The parenthesis at –2 indicates that –2 is not part of the graph. (

Graph Intervals on a Number Line Example 2 Graph 3 > x. The 3 > x means the same as x < 3. The inequality symbol continues to point to the lesser value. The graph of x < 3 in interval notation is written as )

Use the Addition Property of Inequality For any real numbers A, B, and C, the inequalities A < B and A + C < B + C have exactly the same solutions. In words, the same number may be added to each side of an inequality without changing the solutions. Note As with the addition property of equality, the same number may be subtracted from each side of an inequality.

Use the Addition Property of Inequality Example 3 Solve 5 + 6x ≤ 5x + 8, and graph the solution set. 5 + 6x ≤ 5x + 8 A graph of the solution set is –5x –5x 5 + x ≤ 8 ] –5 –5 x ≤ 3

Use the Multiplication Property of Inequality For any real numbers A, B, and C (C ≠ 0), 1. If C is positive, then the inequalities A < B and AC < BC have exactly the same solutions; 2. If C is negative, then the inequalities A < B and AC > BC have exactly the same solutions. In words, each side of an inequality may be multi- plied by the same positive number without changing the solutions. If the multiplier is negative, we must reverse the direction of the inequality symbol.

Use the Multiplication Property of Inequality Note As with the multiplication property of equality, the same nonzero number may be divided into each side. If the divisor is negative, we must reverse the direction of the inequality. This property of inequality holds for any type of inequality (<, >, ≤, and ≥).

Using the Mulitiplication Property of Inequality Example 4 Solve 6y > 12, and graph the solution set. 6y > 12 y > 2 A graph of the solution set is (

Solve Inequalities Using Both Properties of Inequality Solving a Linear Inequality Step 1 Simplify each side separately. Use the distributive property to clear parentheses and combine terms on each side as needed. Step 2 Isolate the variable term on one side. Use the addition property to get all terms with variables on one side of the inequality and all numbers on the other side. Step 3 Isolate the variable. Use the multiplication property to write the inequality in the form x < c or x > c.

Solve Inequalities Using Both Properties of Inequality Example 7 Solve –2(z + 3) – 5z ≤ 4(z – 1) + 9. Graph the solution set. –2(z + 3) – 5z ≤ 4(z – 1) + 9 A graph of the solution set is –2z – 6 – 5z ≤ 4z – 4 + 9 [ –7z – 6 ≤ 4z + 5 –4z –4z –11z – 6 ≤ 5 +6 + 6 Reverse the direction of the inequality symbol when dividing each side by a negative number. –11z ≤ 11 z ≥ –1

Use Inequalities to Solve Applied Problems Phrase Example Inequality Is greater than A number is greater than 4 x > 4 Is less than A number is less than –12 x < –12 Is at least A number is at least 6 x ≥ 6 Is at most A number is at most 8 x ≤ 8

Solve Linear Inequalities With Three Parts Example 9 Graph –1 < x ≤ 2. The statement –1 < x ≤ 2 is read “–1 is less than x and x is less than or equal to 2.” We graph the solutions to this inequality by placing a parenthesis at –1 (because –1 is not part of the graph) and a bracket at 2 (because 2 is part of the graph), then drawing a line segment between the two. Notice that the graph includes all points between –1 and 2 and includes 2 as well. ( ]

Solve Linear Inequalities With Three Parts Example 10 Solve [ )

Use Inequalities to Solve Applied Problems Example 11 Brent has test grades of 86, 88, and 78 on his first three tests in geometry. If he wants an average of at least 80 after his fourth test, what are the possible scores he can make on that test? Let x = Brent’s score on his fourth test. To find his average after four tests, add the test scores and divide by 4. 252 + x ≥ 320 –252 –252 x ≥ 68 Brent must score 68 or more on the fourth test to have an average of at least 80.