An inequality is a statement involving more than one expression and/or number. When two expressions are set greater than or less than one another, you want to determine for what numbers the statement is true. Inequalities can also involve several statements, one greater than the next, greater than the next, and so on. Solving these statements involves treating each section exactly the same and using the rules for dealing with inequalities.
In short, here's what you'll be doing in this chapter:
As you zip through the problems in this chapter, keep the following in mind:
716–719 Perform the indicated operation on the inequalities.
716. Starting with 7 > 3, add −2 to each side, and then multiply each side by −4.
717. Starting with −4 < 1, multiply each side by −2, and then subtract 3 from each side.
718. Starting with −6 ≤ 6, divide each side by −3, and then add 3 to each side of the equation.
719. Starting with 0 ≥ −4, add 3 to each side of the equation and then multiply each side by −1.
720–723 Change the inequality notation to interval notation.
720. −3 ≤ x < 2
721. 0 ≤ x ≤ 4
722. x > −3
723. x ≤ 7
724–727 Change the interval notation to inequality notation.
724. [−6, ∞)
725. (−∞, −2)
727. (2, 3)
728–733 Solve each linear inequality for the values of the variable.
728. 2x − 5 < 3
729. 3x − 2 ≥ 4x + 3
730. −3(x + 7) ≤ 2x + 9
731.
732.
733.
734–737 Solve each compound inequality
734. −5 ≤ 3x + 1 < 7
735. −4 < 6 − 5x < 11
736.
737. −15 < −3(3 − 2x) < −9
738–745 Solve each quadratic inequality using a number line.
738. (x − 3)(x + 4) < 0
739. (2x + 5)(x + 8) ≥ 0
740. x2 − 8x − 9 ≤ 0
742. 48 − x2 > −2x
743. 36 − x2 ≤ 0
744. 5x2 < 15x
745. x2 + 4x + 4 ≥ 0
746–753 Solve each nonlinear inequality using a number line.
746. x(x + 3)(x − 2) > 0
747. (x + 1)2(x + 5)(x − 7) ≤ 0
748. x3 + x2 − 36x − 36 ≥ 0
749. x3 − 2x2 + x < 0
750.
751.
752.
753.
754–761 Solve the absolute value inequalities by rewriting the statements.
754. |3x + 2| ≥ 7
755. |4 − x| < 6
757.
758. |x + 4| − 5 > 3
759. |5 − 2x| + 4 ≤ 7
760. 2|x − 5| − 4 ≥ −2
761.
762–765 Solve the complex inequalities.
762. −4 < 3x + 2 ≤ 2x + 3
763.
764. −5 < 4x − 1 < 6x + 7
765. x + 1 ≤ 3x + 5 < 8