OBJECTIVE 2 Use the addition property of inequality.Consider the true
inequality If 4 is added to each side, the result is
Add 4.
Truealso a true sentence. This example suggests the addition property of inequality.
6 6 9,
2 + 465 + 4
26 5.
SECTION 2.8 Solving Linear Inequalities 153
CHECK Related equation
Let
Multiply.✓ True
A true statement results, so is indeed the “boundary” point. Next we test a num-
ber other than from the interval We choose 0.
CHECK Original inequality
Let7 Ú- 5 ✓ True
7 + 3102 Ú x=0.
?2102 - 5
7 + 3 xÚ 2 x- 5
- 12 3 - 12, q 2.
- 12
- 29 =- 29
7 - 36 - 24 - 5
7 + 31 - 122 21 - 122 - 5 x=-12.
7 + 3 x= 2 x- 5
Addition Property of InequalityIf A, B, and Crepresent real numbers, then the inequalities
and
have exactly the same solutions.
That is, the same number may be added to each side of an inequality without
changing the solutions.
A<B AC<BC
Using the Addition Property of InequalitySolve and graph the solution set.
Subtract 2x.
Combine like terms.
Subtract 7.
Combine like terms.The solution set is 3 - 12, q 2 .Its graph is shown in FIGURE 21.
xÚ- 12
7 +x- 7 Ú- 5 - 7
7 +xÚ- 5
7 + 3 x- 2 xÚ 2 x- 5 - 2 x
7 + 3 xÚ 2 x- 5
7 + 3 xÚ 2 x-5,
EXAMPLE 2
–13 –12–11–10–9–8–7–6–5–4–3–2–1 0
FIGURE 21 NOW TRYNOTEBecause an inequality has many solutions, we cannot check all of them by
substitution as we did with the single solution of an equation. To check the solutions
in Example 2,we first substitute - 12 for xin the related equation.
The checks confirm that solutions to the inequality are in the interval Any
number “outside” the interval that is, any number in will give
a false statement when tested. (Try this.)
3 - 12, q 2 , 1 - q, - 122 ,
3 - 12, q 2.
NOW TRY
EXERCISE 2
Solve the inequality, and
graph the solution set.
5 + 5 xÚ 4 x+ 3–4–3–2–1 012NOW TRY ANSWER
- 3 - 2, q 2
0 is easy to
substitute.