SalasSV_01_03

12 CHAPTER 1 PRECALCULUS REVIEW 1.3 REVIEW OF INEQUALITIES Allour wo rk wit h ine qua lit iesis bas ed on the order proper tie s of the rea l number s gi ven in Section 1.2. In this section we work with the type of inequalities that abound in calculus, namely inequalities that involve a variabl e. To solve an equation in x is to ﬁnd the set of numbers x for which the equation holds. T o solve an inequality in x is to ﬁnd the set of numbers x for which the inequality holds. The way we solve an inequality is very similar to the way we solve an equation, but there is one important difference. We can maintain an inequality by adding the same number to both sides, or by subtracting the same number from both sides, or by multiplying or dividing both sides by the same positive number. But if we multiply or divide by a negative number, then the inequality is reversed : x − 2 < 4 gi ves x < 6, x + 2 < 4 gi ves x < 2, 1 2 x < 4 gi ve s x < 8, but − 1 2 x < 4 gi v es x > −8. ↑ note, the inequality is reversed Example 1 Solve the inequality −3(4 − x) ≤ 12. SOLUTION Multiplying both sides of the inequality by − 1 3 , we have 4 − x ≥−4. (the inequality has been reversed) Subtracting 4, gives −x ≥−8. To isolate x, we multiply by −1. This gives x ≤ 8. (the inequality has been reversed again) The solution set is the interval (−∞, 8] . 8 There are generally several ways to solve a given inequality. For example, the last inequality could have been solved this way: −3(4 − x) ≤ 12, −12 + 3x ≤ 12, 3x ≤ 24, (we added 12) x ≤ 8. (we divided by 3) An approach to solving a quadratic inequality is to factor the quadratic, if possible. Example 2 Solve the inequality x 2 − 4x + 3 > 0. SOLUTION Factoring the quadratic, we obtain

SalasSV_01_03

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