1. Section 2.7 Solving Inequalities

2. Objectives Determine whether a number is a solution of an inequality Graph solution sets and use interval notation Solve linear inequalities Solve compound inequalities Solve inequality applications

3. Objective 1: Determine Whether a Number is a Solution of an Inequality An inequality is a statement that contains one or more of the following symbols.  < is less than  ≤ is less than or equal to  > is greater than  ≥ is greater than or equal to  ≠ is not equal to An inequality can be true, false, or neither true nor false.  An inequality that contains a variable can be made true or false depending on the number that is substituted for the variable.  A number that makes an inequality true is called a solution of the inequality, and we say that the number satisfies the inequality.

4. Objective 1: Determine Whether a Number is a Solution of an Inequality A linear inequality in one variable can be written in one of the following forms where a, b, and c are real numbers and a ≠ 0.  ax + b > c  ax + b ≥ c  ax + b < c  ax + b ≤ c

5. EXAMPLE 1 Is 9 a solution of 2x + 4 ≤ 21?

6. Objective 2: Graph Solution Sets and Use Interval Notation Graph: x ≤ 2 EXAMPLE 2

7. Objective 3: Solve Linear Inequalities To solve an inequality means to find all values of the variable that make the inequality true. As with equations, there are properties that we can use to solve inequalities.  Addition and Subtraction Properties of Inequality: Adding the same number to, or subtracting the same number from, both sides of an inequality does not change its solutions. For any real numbers a, b, and c, if a < b, then a + c < b + c. if a < b, then a – c < b – c. Similar statements can be made for the symbols ≤, >, and ≥.  After applying one of these properties, the resulting inequality is equivalent to the original one. Equivalent inequalities have the same solution set. Like equations, inequalities are solved by isolating the variable on one side.

8. EXAMPLE 3 Solve x + 3 > 2. Write the solution set in interval notation and graph it.

9. Objective 4: Solve Compound Inequalities Two inequalities can be combined into a compound inequality to show that an expression lies between two fixed values.  For example, –2 < x < 3 is a combination of –2 < x and x < 3. It indicates that x is greater than –2 and that x is also less than 3. The solution set of –2 < x < 3 consists of all numbers that lie between –2 and 3, and we write it as (–2, 3).  The graph of the compound inequality is shown below.

10. EXAMPLE 9 Graph –4 ≤ x < 0 and write the solution set in interval notation.

11. Objective 5: Solve Inequality Applications When solving problems, phrases such as “not more than,” or “should exceed” suggest that the problem involves an inequality rather than an equation.

12. EXAMPLE 11 A student has scores of 72%, 74%, and 78% on three exams. What percent score does he need on the last exam to earn a grade of no less than B (80%)? Grades