Chapter 4 Inequalities and Problem Solving

1
Chapter 4Inequalities and Problem Solving
2
4.1
Solving Linear Inequalities
3
Linear Inequalities p
241
Intervals on the Real Number Line Intervals on the Real Number Line Intervals on the Real Number Line
Let a and b be real numbers such that a lt b. Let a and b be real numbers such that a lt b. Let a and b be real numbers such that a lt b.
Interval Notation Set-Builder Notation Graph
(a,b) xa lt x lt b
a,b xa x b
a,b) xa x lt b
(a,b xa lt x b
(a, ) xx gt a
a, ) xx a
(- ,b) xx lt b
(- ,b xx b
(- , ) xx is a real number
(
)
a
b


a
b
)

a
b
The parentheses in the graph and in interval
notation indicate that a and b, the endpoints,
are excluded from the interval.
(

X is greater than a (altx)andx is less than b
(xltb)
a
b
(
a

a
)
b

b
Blitzer, Intermediate Algebra, 5e Slide 3
Section 4.1
4
Linear Inequalities p
241
Intervals on the Real Number Line Intervals on the Real Number Line Intervals on the Real Number Line
Let a and b be real numbers such that a lt b. Let a and b be real numbers such that a lt b. Let a and b be real numbers such that a lt b.
Interval Notation Set-Builder Notation Graph
(a,b) xa lt x lt b
a,b xa x b
a,b) xa x lt b
(a,b xa lt x b
(a, ) xx gt a
a, ) xx a
(- ,b) xx lt b
(- ,b xx b
(- , ) xx is a real number
(
)
a
b


a
b
)

a
b
(

a
b
(
a
The square brackets in the graph and in interval
notation indicate that a and b, the endpoints,
are included in the interval.
X is greater than or equal to a a xandx
is less than or equal to b (x b)

a
)
b

b
Blitzer, Intermediate Algebra, 5e Slide 4
Section 4.1
5
Linear Inequalities p
241
Intervals on the Real Number Line Intervals on the Real Number Line Intervals on the Real Number Line
Let a and b be real numbers such that a lt b. Let a and b be real numbers such that a lt b. Let a and b be real numbers such that a lt b.
Interval Notation Set-Builder Notation Graph
(a,b) xa lt x lt b
a,b xa x b
a,b) xa x lt b
(a,b xa lt x b
(a, ) xx gt a
a, ) xx a
(- ,b) xx lt b
(- ,b xx b
(- , ) xx is a real number
(
)
a
b


a
b
)

a
b
(

a
b
(
a

a
)
b

b
Blitzer, Intermediate Algebra, 5e Slide 5
Section 4.1
6
Linear Inequalities p 242
EXAMPLE
Graph the solution of each inequality on a number
line and then express the solutions in
set-builder notation and interval notation.
SOLUTION
(a) The solution to x gt -4 is all real numbers
that are greater than -4. They are graphed on a
number line by shading all points to the right of
-4. The parenthesis at -4 indicates that -4 is
not a solution, but numbers such as -3.9999 and
-3.3 are. The arrow shows that the graph extends
indefinitely to the right.
xx gt -4 (-4, )
(
-4
Blitzer, Intermediate Algebra, 5e Slide 6
Section 4.1
7
Linear Inequalities p 242
CONTINUED
(b) The solution to is all real
numbers that are greater than -2 and less than 7.
They are graphed on a number line by shading all
points that are to the right of -2 and to the
left of 7. The bracket at -2 indicates that -2
is part of the solution. The parenthesis at 7
indicates that 7 is not part of the solution.
x-2 x lt 7 -2,7)
)

-2
7
Blitzer, Intermediate Algebra, 5e Slide 7
Section 4.1
8
Linear Inequalities p 242
Check Point 1

)

)
)
Blitzer, Intermediate Algebra, 5e Slide 8
Section 4.1
9
Inequalities p
243
Properties of Inequalities Properties of Inequalities Properties of Inequalities
Property The Property in Words Example
The Addition Property of Inequality If a lt b, then a c lt b c If a lt b, then a c lt b - c If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one. 2x 3 lt 7 Subtract 3 2x 3 3 lt 7 3 Simplify 2x lt 4
Blitzer, Intermediate Algebra, 5e Slide 9
Section 4.1
10
Inequalities
p 243
CONTINUED
Properties of Inequalities Properties of Inequalities Properties of Inequalities
Property The Property in Words Example
The Positive Multiplication Property of Inequality If a lt b and c is positive, then ac lt bc If a lt b and c is positive, then If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one. 2x lt 4 Divide by 2 Simplify x lt 2
Blitzer, Intermediate Algebra, 5e Slide 10
Section 4.1
11
Inequalities
p 243
CONTINUED
Properties of Inequalities Properties of Inequalities Properties of Inequalities
Property The Property in Words Example
The Negative Multiplication Property If a lt b and c is negative, then ac gt bc If a lt b and c is negative, then If we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the resulting inequality is equivalent to the original one. -4x lt 20 Divide by -4 and reverse the sense of the inequality Simplify x gt -5
Note that when you multiply both sides of an
inequality by a negative, you must reverse the
direction of the inequality symbol
Blitzer, Intermediate Algebra, 5e Slide 11
Section 4.1
12
Inequalities p
243
Solving a Linear Inequality
1) Simplify the algebraic expression on each side.
2) Use the addition property of inequality to collect all the variable terms on one side and all the constant terms on the other side.
3) Use the multiplication property of inequality to isolate the variable and solve. Reverse the sense of the inequality when multiplying or dividing both sides by a negative number.
4) Express the solution set in set-builder or interval notation and graph the solution set on a number line.
Blitzer, Intermediate Algebra, 5e Slide 12
Section 4.1
13
Linear Inequalities p 244
EXAMPLE
Solve the linear inequality. Then graph the
solution set on a number line.
SOLUTION
1) Simplify each side.
Distribute
2) Collect variable terms on one side and
constant terms on the other side.
Add 5x to both sides
Add 1 to both sides
Blitzer, Intermediate Algebra, 5e Slide 13
Section 4.1
14
Linear Inequalities
CONTINUED
3) Isolate the variable and solve.
Divide both sides by 8
4) Express the solution set in set-builder or
interval notation and graph the set on a number
line.

-1 0 1 2 3 4 5
6
xx 2 2, )
Blitzer, Intermediate Algebra, 5e Slide 14
Section 4.1
15
Linear Inequalities p 244 -
245
Check Point 2 (complete on board)
Check Point 3 (complete on board)
Blitzer, Intermediate Algebra, 5e Slide 15
Section 4.1
16
Linear Inequalities p 246
EXAMPLE
Solve the linear inequality. Then graph the
solution set on a number line.
SOLUTION
First we need to eliminate the denominators.
Multiply by LCD 10
Distribute
Blitzer, Intermediate Algebra, 5e Slide 16
Section 4.1
17
Linear Inequalities
CONTINUED
1 2
1
1 1
1
1) Simplify each side. Because each side is
already simplified, we can skip this step.
2) Collect variable terms on one side and
constant terms on the other side.
Add x to both sides
Subtract 10 from both sides
Blitzer, Intermediate Algebra, 5e Slide 17
Section 4.1
18
Linear Inequalities
CONTINUED
3) Isolate the variable and solve.
Divide both sides by 4
4) Express the solution set in set-builder or
interval notation and graph the set on a number
line.

-6 -5 -4 -3 -2 -1 0
1 2
xx -2 -2, )
Blitzer, Intermediate Algebra, 5e Slide 18
Section 4.1
19
Linear Inequalities p 247
Check Point 4 (Complete on board)
Blitzer, Intermediate Algebra, 5e Slide 19
Section 4.1
20
Linear Inequalities
EXAMPLE OF NO SOLUTION
Solve the linear inequality.
5x lt 5(x 3)
SOLUTION
5x lt 5(x 3)
Distribute
5x lt 5x 15
Subtract 5x from both sides
0 lt 15
Since the result is an obviously false statement,
there is no solution.
Blitzer, Intermediate Algebra, 5e Slide 20
Section 4.1
21
Linear Inequalities p 248
English Sentences and Inequalities English Sentences and Inequalities
English Sentence Inequality
x is at least 5.
x is at most 5.
x is between 5 and 7.
x is no more than 5.
x is no less than 5.
Blitzer, Intermediate Algebra, 5e Slide 21
Section 4.1
22
Linear Inequalities
EXAMPLE
You are choosing between two long-distance
telephone plans. Plan A has a monthly fee of 15
with a charge of 0.08 per minute for all
long-distance calls. Plan B has a monthly fee of
3 with a charge of 0.12 per minute for all
long-distance calls. How many minutes of
long-distance calls in a month make plan A the
better deal?
SOLUTION
1) Let x represent one of the quantities. We are
looking for the number of minutes of
long-distance calls that make Plan A the better
deal. Thus,
Let x the number of minutes of long-distance
phone calls.
Blitzer, Intermediate Algebra, 5e Slide 22
Section 4.1
23
Linear Inequalities
CONTINUED
2) Represent other quantities in terms of x. We
are not asked to find another quantity, so we can
skip this step.
3) Write an inequality in x that describes the
conditions. Plan A is a better deal than Plan B
if the monthly cost of Plan A is less than the
monthly cost of Plan B. The inequality that
represents this is the following, with the
information for Plan A on the left side and the
information for Plan B on the right side of the
inequality.
15 0.08x lt 3 0.12x
Plan A is less than Plan B
Blitzer, Intermediate Algebra, 5e Slide 23
Section 4.1
24
Linear Inequalities
CONTINUED
4) Solve the inequality and answer the question.
15 0.08x lt 3 0.12x
Subtract 3 from both sides
12 0.08x lt 0.12x
Subtract 0.08x from both sides
12 lt 0.04x
Divide both sides by 0.04
300 lt x
Therefore, Plan A costs less than Plan B when x gt
300 long-distance minutes per month.
Blitzer, Intermediate Algebra, 5e Slide 24
Section 4.1
25
Linear Inequalities
CONTINUED
5) Check the proposed solution in the original
wording of the problem. One way to do this is to
take a long-distance number of minutes greater
than 300 per month to see if Plan A is the better
deal. Suppose that the number of long-distance
minutes we use is 450 in a month.
Cost for Plan A 15 (450)(0.08) 51
Cost for Plan B 3 (450)(0.12) 57
Plan A has a lower monthly cost, making Plan A
the better deal.
Blitzer, Intermediate Algebra, 5e Slide 25
Section 4.1
26
Linear Inequalities
In summary
Solving a linear inequality is similar to solving
a linear equation except for two important
differences When you solve an inequality, your
solution is usually an interval, not a
point. and When you multiply both sides of an
inequality by a negative number, you must
remember to reverse the direction of the
inequality sign that is, POW (point the other
way).
Blitzer, Intermediate Algebra, 5e Slide 26
Section 4.1
27
DONE