168 CHAPTER 1 Equations and Inequalities 1.7 Inequalities ■ Linear Inequalities ■ Three-Part Inequalities ■ Quadratic Inequalities ■ Rational Inequalities An inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to another. As with equations, a value of the variable for which the inequality is true is a solution of the inequality, and the set of all solutions is the solution set of the inequality. Two inequalities with the same solution set are equivalent. Inequalities are solved with the properties of inequality, which are similar to the properties of equality. Properties of Inequality Let a, b, and c represent real numbers. 1. If a *b, then a +c *b +c. 2. If a *b and if c +0, then ac *bc. 3. If a *b and if c *0, then ac +bc. Replacing 6 with 7, …, or Ú results in similar properties. (Restrictions on c remain the same.) Linear Inequalities The definition of a linear inequality is similar to the definition of a linear equation. Linear Inequality in One Variable A linear inequality in one variable is an inequality that can be written in the form ax +b +0,* where a and b are real numbers and a≠0. * The symbol 7 can be replaced with 6, …, or Ú. EXAMPLE 1 Solving a Linear Inequality Solve -3x + 5 7 -7. SOLUTION -3x + 5 7 -7 -3x + 5 - 5 7 -7 - 5 Subtract 5. -3x 7 -12 Combine like terms. -3x -3 6 -12 -3 Divide by -3. Reverse the direction of the inequality symbol when multiplying or dividing by a negative number. x 64 Thus, the original inequality -3x + 5 7 -7 is satisfied by any real number less than 4. The solution set can be written 5x x 646. Don’t forget to reverse the inequality symbol here. NOTE Multiplication may be replaced by division in Properties 2 and 3. Always remember to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.
RkJQdWJsaXNoZXIy NjM5ODQ=