MA 3972-MA-Book May 8, 2018 13:46
Review of Precalculus 51[−10, 10] by [−10, 10]
Figure 5.2-1Inequalities and the Real Number Line
Properties of Inequalities
Leta,b,c,d, andkbe real numbers:
- Ifa<bandb<c, thena<c.For example,− 7 <2 and 2< 5 ⇒− 7 <5.
- Ifa<bandc<d, thena+c<b+d.For example, 5<7 and 3< 6 ⇒ 5 + 3
< 7 +6. - Ifa
0, thenak<bk.For example, 3<5 and 2> 0 ⇒3(2)<5(2). - Ifa<bandk<0, thenak>bk.For example, 3 <5 and− 2 < 0 ⇒ 3(−2)
5(−2).
Example 1
Solve the inequality 6− 2 x≤18 and sketch the solution on the real number line.
Solving the inequality 6− 2 x≤18 gives
− 2 x≤ 12− (^60)
[
x≥− 6
Therefore, the solution set is the interval [−6,∞) or, by expressing the solution set in set
notation,{x|x≥− 6 }.
Example 2
Solve the double inequality− 15 ≤ 3 x+ 6 <9 and sketch the solution on the real number
line.
Solving the double inequality− 15 ≤ 3 x+ 6 <9 gives
− 21 ≤ 3 x< (^3)
1
)[
− 7 ≤x< 1 −^7
Therefore, the solution set is the interval [−7, 1) or, by expressing the solution in set
notation:{x|− 7 ≤x< 1 }.