MATH 125 Section 2.7 - {2020} | MATH125 Section 2.7 _ Graded A

Solve Linear Inequalities Learning Objectives By the end of this section, you will be able to: Graph inequalities on the number line Solve inequalities using the Subtraction and Addition Properties of inequality Solve inequalities using the Division and Multiplication Properties of inequality Solve inequalities that require simplification Translate to an inequality and solve Be Prepared! Before you get started, take this readiness quiz. 1. Translate from algebra to English: 15 > x . If you missed this problem, review Example 1.12. 2. Solve: n − 9 = −42. If you missed this problem, review Example 2.3. 3. Solve: −5p = −23. If you missed this problem, review Example 2.13. 4. Solve: 3a − 12 = 7a − 20. If you missed this problem, review Example 2.34. Graph Inequalities on the Number Line Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. What about the solution of an inequality? What number would make the inequality x > 3 true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality x > 3 . We show the solutions to the inequality x > 3 on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of x > 3 is shown in Figure 2.7. Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction. Figure 2.7 The inequality x > 3 is graphed on this number line. The graph of the inequality x ≥ 3 is very much like the graph of x > 3 , but now we need to show that 3 is a solution, too. We do that by putting a bracket at x = 3 , as shown in Figure 2.8. Figure 2.8 The inequality x ≥ 3 is graphed on this number line. Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included. EXAMPLE 2.66 Graph on the number line: 270 Chapter 2 Solving Linear Equations and Inequalities This OpenStax book is available for free at Solution ⓐ x ≤ 1 This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at x = 1 to show that it is included. ⓑ x < 5 This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at x = 5 to show it is not included. ⓒ x > − 1 This means all numbers greater than −1 , but not including −1 . We shade in all the numbers on the number line to the right of −1 , then put a parenthesis at x = −1 to show it is not included. TRY IT : : 2.131 Graph on the number line: ⓐ x ≤ − 1 ⓑ x > 2 ⓒ x < 3 TRY IT : : 2.132 Graph on the number line: ⓐ x > − 2 ⓑ x < − 3 ⓒ x ≥ −1 We can also represent inequalities using interval notation. As we saw above, the inequality x > 3 means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express x > 3 as (3, ∞). The symbol ∞ is read as ‘infinity’. It is not an actual number. Figure 2.9 shows both the number line and the interval notation. Figure 2.9 The inequality x > 3 is graphed on this number line and written in interval notation. The inequality x ≤ 1 means all numbers less than or equal to 1. There is no lower end to those numbers. We write x ≤ 1 in interval notation as (−∞, 1] . The symbol −∞ is read as ‘negative infinity’. Figure 2.10 shows both the number line and interval notation. Figure 2.10 The inequality x ≤ 1 is graphed on this number line and written in interval notation. Chapter 2 Solving Linear Equations and Inequalities 271Inequalities, Number Lines, and Interval Notation Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure 2.11. Figure 2.11 The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals. EXAMPLE 2.67 Graph on the number line and write in interval notation. ⓐ x ≥ −3 ⓑ x < 2.5 ⓒ x ≤ − 3 5 Solution ⓐ Shade to the right of −3 , and put a bracket at −3 . Write in interval notation. ⓑ Shade to the left of 2.5 , and put a parenthesis at 2.5 . Write in interval notation. ⓒ Shade to the left of −3 5 , and put a bracket at −3 5 . Write in interval notation. TRY IT : : 2.133 Graph on the number line and write in interval notation: ⓐ x > 2 ⓑ x ≤ − 1.5 ⓒ x ≥ 3 4 272 Chapter 2 Solving Linear Equations and Inequalities This OpenStax book is available for free at ⓐ x ≤ − 4 ⓑ x ≥ 0.5 ⓒ x < − 2 3 Solve Inequalities using the Subtraction and Addition Properties of Inequality The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal. Properties of Equality Subtraction Property of Equality Addition Property of Equality For any numbers a, b, and c, For any numbers a, b, and c, if a = b, then a − c = b − c. if a = b, then a + c = b + c. Similar properties hold true for inequalities. For example, we know that −4 is less than 2. If we subtract 5 from both quantities, is the left side still less than the right side? We get −9 on the left and −3 on the right. And we know −9 is less than −3. The inequality sign stayed the same. Similarly we could show that the inequality also stays the same for addition. This leads us to the Subtraction and Addition Properties of Inequality. Properties of Inequality Subtraction Property of Inequality Addition Property of Inequality For any numbers a, b, and c, For any numbers a, b, and c, if a < b then a − c < b − c. if a > b then a − c > b − c. if a < b then a + c < b + c. if a > b then a + c > b + c. We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality x + 5 > 9 , the steps would look like this: x + 5 > 9 Subtract 5 from both sides to isolate x. x + 5 − 5 > 9 − 5 Simplify. x > 4 Any number greater than 4 is a solution to this inequality. EXAMPLE 2.68 Solve the inequality n − 1 2 ≤ 5 8 , graph the solution on the number line, and write the solution in interval notation. Chapter 2 Solving Linear Equations and Inequalities 273Solution Add 1 2 to both sides of the inequality. Simplify. Graph the solution on the number line. Write the solution in interval notation.