1. LIAL HORNSBY SCHNEIDER
COLLEGE ALGEBRA
LIAL
HORNSBY
SCHNEIDER

2. Inequalities 1.7 Linear Inequalities Three-Part Inequalities
Quadratic Inequalities
Rational Inequalities

3. Properties of Inequality
For real numbers a, b and c:
If a < b, then a + c < b + c,
If a < b and if c > 0, then ac < bc,
If a < b and if c < 0, then ac > bc.
Replacing < with >, ≤ , or ≥ results in similar properties.

4. Motion Problems
Note Multiplication may be replaced by division in properties 2 and 3.
Always remember to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

5. Linear Inequality in One Variable
A linear inequality in one variable is an inequality that can be written in the form
where a and b are real numbers with
a ≠ 0. (Any of the symbols ≥, <, or ≤ may also be used.)

6. Don’t forget to reverse the symbol here.
SOLVING A LINEAR INEQUALITY
Example 1
Solve
Solution
Subtract 5.
Divide by – 3; reverse direction of the inequality symbol when multiplying or dividing by a negative number.
Don’t forget to reverse the symbol here.

7. Set Interval Notation Graph
Type of Interval
Set
Interval Notation
Graph
Open interval
{x  x > a}
{x  a < x < b}
{x  x < b}
(a, )
(a, b)
(– , b)
Other interval
{x  x ≥ a}
{x  a < x ≤ b}
{x  a ≤ x < b}
{x  x≤ b}
[a, )
(a, b]
[a, b)
(– , b] ( a ( a b ( b [ a ( ] a b [ ) a b ] b

8. Set Interval Notation Graph
Type of Interval
Set
Interval Notation
Graph
Closed interval
{xa ≤ x ≤ b}
[a, b]
Disjoint interval
{xx < a or x > b}
(– , a) (b, )
All real numbers
{xx is a real numbers}
(– , ) [ ] a b ( a b

9. SOLVING A LINEAR INEQUALITY
Example 2
Solve 4 – 3x ≤ 7 + 2x. Give the solution set in interval notation and graph it.
Solution
Subtract 4.
Subtract 2x.

10. SOLVING A LINEAR INEQUALITY
Example 2
Solve 4 – 3x ≤ 7 + 2x. Give the solution set in interval notation and graph it.
Solution
Divide by –5; reverse the direction of the inequality symbol. [

11. Solve – 2 < 5 + 3x < 20. Solution
SOLVING A THREE-PART INEQUALITY
Example 3
Solve – 2 < 5 + 3x < 20.
Solution
Subtract 5 from each part.
Divide each part by 3.

12. If the revenue and cost of a certain product are given by
FINDING THE BREAK-EVEN POINT
Example 4
If the revenue and cost of a certain product are given by
where x is the number of units produced and sold, at what production level does R at least equal C?
Solution Set R ≥ C and solve for x.

13. Solution Set R ≥ C and solve for x.
FINDING THE BREAK-EVEN POINT
Example 4
Solution Set R ≥ C and solve for x.
Substitute.
Subtract 2x.
Divide by 2.
The break-even point is at x = This product will at least break even only if the number of units produced and sold is in the interval [500, ).

14. Quadratic Inequalities
A quadratic inequality is an inequality that can be written in the form
for real numbers a, b, and c with a ≠ 0. (The symbol < can be replaced with >, ≤, or ≥.)

15. Solving a Quadratic Inequality
Step 1 Solve the corresponding quadratic equation.
Step 2 Identify the intervals determined by the solutions of the equation.
Step 3 Use a test value from each interval to determine which intervals form the solution set.

16. Step 1 Find the values of x that satisfy x2 – x – 12 = 0
SOLVING A QUADRATIC INEQULITY
Example 5
Solve
Solution
Step 1 Find the values of x that satisfy
x2 – x – 12 = 0
Corresponding quadratic equation
Factor. or
Zero-factor property

17. Solve Solution Step 1 or or SOLVING A QUADRATIC INEQULITY Example 5
Zero-factor property or

18. SOLVING A QUADRATIC INEQULITY
Example 5
Step 2 The two numbers – 3 and 4 divide the number line into three intervals. If a value in Interval A makes the polynomial negative, then all values in Interval A will make the polynomial negative.
– 3 4
Interval A
Interval B
Interval C
(–, – 3)
(– 3, 4)
(4, )

19. Step3 Choose a test value from each interval.
SOLVING A QUADRATIC INEQULITY
Example 5
Step3 Choose a test value from each interval.
Interval
Test Value
Is x2 – x – 12< 0
True or False?
A: (– , – 3)
– 4
(– 4)2 – 4(– 4) – 12 < ?
8 < False
B: (– 3, 4)
02 – 0 – 12 < ?
– 12 < True
C: (4, ) 5
52 – 5 – 12 < ?
Since the values in Interval B make the inequality true, the solution set is (– 3, 4).

20. Step 1 Find the values of x that satisfy this equation.
SOLVING QUADRATIC INEQULAITY
Example 6
Solve
Solution
Step 1 Find the values of x that satisfy this equation.
Corresponding quadratic equation
Factor. or
Zero-factor property

21. Step 1 Find the values of x that satisfy this equation.
SOLVING QUADRATIC INEQULAITY
Example 6
Solve
Solution
Step 1 Find the values of x that satisfy this equation. or
Zero-factor property or

22. Step 2 The values form the intervals on the number line.
SOLVING QUADRATIC INEQULAITY
Example 6
Solve
Solution
Step 2 The values form the intervals on the number line.
– 4
3/2
Interval A
Interval B
Interval C
(–, – 4)
(– 4, 3/2)
(3/2, )

23. Step3 Choose a test value from each interval.
SOLVING QUADRATIC INEQULAITY
Example 6
Step3 Choose a test value from each interval.
Interval
Test Value
Is 2x2 + 5x – 12 ≥ 0
True or False?
A: (– , – 4)
– 5
2(– 4)2 +5(– 5) – 12 ≥ 0 ?
13 ≥ 0 True
B: (4, 3/2)
2(0)2 +5(0) – 12 ≥ 0 ?
– 12 ≥ 0 False
C: (3/2, ) 2
2(2)2 + 5(2) – 12 ≥ 0 ?
6 ≥ True
The values in Intervals A and B make the inequality true, the solution set is the union of the intervals.

24. Inequalities
Note Inequalities that use the symbols < and > are called strict inequalities; ≤ and ≥ are used in nonstrict inequalities. The solutions of the equation in Example 5 were not included in the solution set since the inequality was a strict inequality. In Example 6, the solutions of the equation were included in the solution set because of the nonstrict inequality.

25. When will the projectile be greater than 80 ft above ground level?
SOLVING A PROBLEM INVOLVING THE HEIGHT OF A PROJECTILE
Example 7
If a projectile is launched from ground level with an initial velocity of 96 ft per sec, its height in feet t seconds after launching is s feet, where
When will the projectile be greater than 80 ft above ground level?

26. Reverse direction of the inequality symbol.
SOLVING A PROBLEM INVOLVING THE HEIGHT OF A PROJECTILE
Example 7
Solution
Set s greater than 80.
Subtract 80.
Divide by – 16.
Reverse direction of the inequality symbol.
Now solve the corresponding equation.

27. Solution or SOLVING A PROBLEM INVOLVING THE HEIGHT OF A PROJECTILE
Example 7
Solution
Factor. or
Zero-factor property

28. SOLVING A PROBLEM INVOLVING THE HEIGHT OF A PROJECTILE
Example 7
Solution
Interval C 1 5
Interval A
Interval B
(– , 1)
(1, 5)
(5, )
Use the procedure of Examples 5 and 6 to determine that values in Interval B, (1, 5) , satisfy the inequality. The projectile is greater than 80 ft above ground level between 1 and 5 sec after it is launched.

29. Solving a Rational Inequality
Step 1 Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side.
Step 2 Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals of the number line to consider.

30. Solving a Rational Inequality
Step 3 Use a test value from each interval to determine which intervals form the solution set.
A value causing the denominator to equal zero will never be included in the solution set. If the inequality is strict, any value causing the numerator to equal zero will be excluded; if nonstrict, any such value will be included.

31. by multiplying both sides by x + 4 to obtain
Caution Solving a rational inequality such as
by multiplying both sides by x + 4 to obtain
5 ≥ x + 4 requires considering two cases, since the sign of x + 4 depends on the value of x. If
x + 4 were negative, then the inequality symbol must be reversed. The procedure described in the next two examples eliminates the need for considering separate cases.

32. Solve Solution Step 1 Example 8 SOLVING A RATIONAL INEQUALITY
Subtract 1 so that 0 is on one side.
Use x + 4 as the common denominator.

33. Note the careful use of parentheses.
SOLVING A RATIONAL INEQUALITY
Example 8
Solve
Solution
Step 1
Use x + 4 as the common denominator.
Note the careful use of parentheses.
Write as a single fraction.
Combine terms in the numerator; be careful with signs.

34. SOLVING A RATIONAL INEQUALITY
Example 8
Solve
Solution
Step 2 The quotient possibly changes sign only where x-values make the numerator or denominator 0. This occurs at or or

35. Solve Solution Step 2 Example 8 SOLVING A RATIONAL INEQUALITY
– 4 1
Interval A
Interval B
Interval C
(–, – 4)
(– 4, 1)
(1, )

36. Step3 Choose test values.
SOLVING A RATIONAL INEQUALITY
Example 8
Step3 Choose test values.
Interval
Test Value
A: (– , – 4)
– 5
B: (– 4, 1)
C: (1, ) 2
The values in the interval (– 4, 1) satisfy the original inequality. The value 1 makes the nonstrict inequality true, so it must be included in the solution set. Since – 4 makes the denominator 0, it must be excluded. The solutions set is (– 4, 2].

37. Caution As suggested by Example 8, be careful with the endpoints of the intervals when solving rational inequalities.

38. Solve Solution SOLVING A RATIONAL INEQULAITY Example 9 Subtract 5.
Common denominator is 3x + 4.
Write as a single fraction.

39. Solve Solution SOLVING A RATIONAL INEQULAITY Example 9
Write as a single fraction.
Distributive property
Be careful with signs.
Combine terms in the numerator.

40. SOLVING A RATIONAL INEQULAITY
Example 9
Solve
Solution
Set the numerator and denominator equal to 0 and solve the resulting equations to get the values of x where sign changes may occur. or or

41. Solve Solution SOLVING A RATIONAL INEQULAITY Example 9 Interval A
Interval B
Interval C

42. Solution SOLVING A RATIONAL INEQULAITY Example 9
Now choose from the intervals and verify that:
– 2 from Interval A makes the inequality true;
– 1.5 from Interval B makes the inequality false;
from Interval C makes the inequality true.
Because of the < symbol, neither endpoint satisfies the inequality, so the solution set is