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Solving Linear Inequalities

  • A mathematical statement that uses one of the following inequality signs is called an inequality.

  • Inequalities can involve variables and are similar to equations, except that the two sides are related by one of the inequality signs instead of the equality sign used in equations.
  • To solve an inequality means to find the set of values of the variable that make the inequality true. This set of values is also known as the solution set of an inequality. Two inequalities that have the same solution set are called equivalent inequalities.
  • The procedure used to solve a linear inequality is similar to that used to solve a linear equation, which is to simplify the inequality by isolating the variable on one side of the inequality, using the following two rules.
    • When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original.
    • When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original.

      Ex. The inequality  can be solved as follows.

           (5 subtracted from both sides)

          (both sides divided by -3, which                                                                                                                                                                                                                                                                                                                                        reverses the direction of the inequality)

      Therefore, the solution set of consists of all real numbers greater than or equal to –4.


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