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Course 2: Inequalities Objectives: •To determine whether a number is a solution of an inequality •To graph inequalities on the number line •To write inequalities Inequalities • An inequality is a mathematical sentence containing >, <, >, <. Inequalities - is less than Words Symbols < - is greater than > -is less than or equal to - is at most < -is greater than or equal to -is at least > Inequalities • Any number that makes an inequality true is a solution of the inequality. • Inequalities have many solutions. • Example: x > 4 • List 4 possible solutions. 4.5, 5, 7, 12.5 Example 2 The solutions are shown by shading a number line. Example: x > 4 3 4 5 6 7 Example 1 Determine whether each number is a solution of x 7. a) 3 yes, because 3 is less than 7 b) -2 yes, because -2 is less than 7 c) 9 no, because 9 is not less than or equal to 7 d) 7 yes, because 7 is equal to 7 1) Graph m > 3 on a number line. 1 2 3 4 5 2) Graph k < -2 on a number line. -3 -2 -1 0 1 3) Graph h > 3 on a number line. 0 1 2 3 4 4) Graph k < -2 on a number line. -3 -2 -1 0 1 Solving One-Step Inequalities by Adding or Subtracting • 1) x + 4 > 8 -4 -4 x>4 Check x + 4 > 8 • Solution: x > 4 • Substitute a value that is greater than 4 for x. 5+4>8 9 > 8 This is a true statement. Graph x > 4 1 2 3 4 5 Solving One-Step Inequalities by Adding or Subtracting • 2) c - 3 < 2 +3 +3 c<5 Check c – 3 < 2 • Solution: c < 5 • Substitute a value that is less than or equal to 5 for c. 5–3<2 2 < 2 This is a true statement. Graph c < 5 on a number line. 2 3 4 5 6 Solving One-Step Inequalities by Adding or Subtracting • 3) d - 4 < -2 +4 +4 d<2 Check d – 4 < -2 • Solution: d < 2 • Substitute a value that is less than 2 for d. 1 – 4 < -2 -3 < -2 This is a true statement. Graph d < -2. -5 -4 -3 -2 -1 Solving One-Step Inequalities by Adding or Subtracting • 4) a - 2 > 6 +2 +2 a>8 Check a - 2 > 6 • Solution: a > 8 • Substitute a value that is greater than or equal to 8 for a. 8-2>6 6 > 6 This is a true statement. Graph a > 8. 5 6 7 8 9 Solving One-Step Inequalities by Adding or Subtracting • 5) p - 7 > 0 +7 +7 p>7 Check p - 7 > 0 • Solution: p > 7 • Substitute a value that is greater than 7 for p. 8-7>0 1 > 0 This is a true statement. Graph p > 7 4 5 6 7 8 Solving One-Step Inequalities by Adding or Subtracting • 6) j + 5 < 2 -5 -5 j < -3 Check j + 5 < 2 • Solution: j < -3 • Substitute a value that is less than or equal to -3 for c. -3 + 5 < 2 2 < 2 This is a true statement. Graph j < -3 on a number line. -5 -4 -3 -2 -1 Review