Download Solve One-Variable Inequalities

Solving and Graphing
Inequalities (5-1, 5-2)
Objective: Solve linear
inequalities by using addition and
subtraction. Solve linear
inequalities by using
multiplication and division.
Solve Inequalities by
Addition
• Addition Property of Inequalities:
– If the same number is added to each
side of a true inequality, the resulting
inequality is also true.
– For all numbers a, b, and c, the
following are true.
1. If a > b, then a + c > b + c.
2. If a < b, then a + c < b + c.
– This property is also true for ≥ and ≤.
Example 1
• Solve c – 12 > 65.
c – 12 > 65
+12 +12
c > 77
{all numbers greater than 77}
Check Your Progress
• Choose the best answer for the
following.
– Solve k – 4 < 10.
A.
B.
C.
D.
k > 14
k < 14
k<6
k>6
k – 4 < 10
+4 +4
Notation
• A more concise way of writing a
solution set is to use set-builder
notation.
• In set-builder notation, the set is
written as {variable | inequality}.
• {k|k < 14} would be read as “the
set of all numbers k such that k is
less than 14”.
Solve Inequalities by
Subtraction
• Subtraction Property of
Inequalities:
– If the same number is subtracted
from each side of a true inequality,
the resulting inequality is also true.
– For all numbers a, b, and c, the
following are true.
1. If a > b, then a – c > b – c.
2. If a < b, then a – c < b – c.
– This property is also true for ≥ and ≤.
Example 2
• Solve the inequality x + 23 < 14.
x + 23 < 14
-23 -23
x < -9
{x|x < -9}
Check Your Progress
• Choose the best answer for the
following.
– Solve the inequality m – 4 ≥ -8.
A.
B.
C.
D.
{m|m ≥ 4}
{m|m ≤ -12}
{m|m ≥ -4}
{m|m ≥ -8}
m – 4 ≥ -8
+4 +4
Graphing
• The solution set can be graphed on a
number line.
• The graph will consist of an endpoint and
shading.
• The endpoint will be a circle for > and <.
• The endpoint will be a dot for ≥ and ≤.
• The shading will be to the right for > and ≥.
• The shading will be to the left for < and ≤.
• Always graph an inequality with the variable
on the left of the inequality sign.
Example 3
• Solve 12n – 4 ≤ 13n. Graph the
solution set.
12n – 4 ≤ 13n
-12n









-12n
-4 ≤ n
n ≥ -4
{n|n ≥ -4}




Check Your Progress
• Choose the best answer for the following.
– Solve 3p – 6 ≥ 4p. Graph the solution.
A. {p|p ≤ -6}









 
B. {p|p ≤ -6}









 
C. {p|p ≥ -6}









 









 
D. {p|p ≥ -6}
3p – 6 ≥ 4p
-3p
-3p
-6 ≥ p
Verbal Problems
• Verbal problems containing phrases like
greater than or less than can be solved by
using inequalities.
• The chart shows some other phrases that
indicate inequalities.
<
>
≤
≥
less than
fewer than
greater than
more than
at most
no more than
less than or equal to
at least
no less than
greater than or equal to
Example 4
• Panya wants to buy season passes to two theme
parks. If one season pass costs $54.99 and Panya has
$100 to spend on both passes, the second season
pass must cost no more than what amount?
– Let c = cost of season pass
c + 54.99 ≤ 100
-54.99 -54.99
c ≤ 45.01
– She can spend up to $45.01 on the second season
pass.
Check Your Progress
• Choose the best answer for the
following.
– Jeremiah is taking two of his friends out
for pancakes. If he spends $17.55 on
their meals and has $26 to spend in
total, Jeremiah’s pancakes must cost
no more than what amount?
A.
B.
C.
D.
$8.15
$8.45
$9.30
$7.85
c + 17.55 ≤ 26
-17.55 -17.55
Solve Inequalities by
Multiplication
• Multiplication Property of Inequalities:
– If you multiply each side of an inequality by a
positive number, then the inequality remains true.
– For any real numbers a and b and any positive
number c, if a > b, then ac > bc. And, if a < b, then
ac < bc.
– If you multiply each side of an inequality by a
negative number, the inequality symbol changes
direction.
– For any real numbers a and b and any negative real
number c, if a > b, then ac < bc. And, if a < b, then
ac > bc.
– This property also hold for inequalities involving ≤
and ≥.
Example 5
• Mateo walks at a rate of ¾ mile per hour. He
knows that it is at least 9 miles to Onyx Lake.
How long will it take Mateo to get there?
Write and solve an inequality to find the time.
4
3
4
3
h9
3
4
h ≥ 12
It will take at least 12 hours.
Check Your Progress
• Choose the best answer for the
following.
– At Midpark High School, 2/3 of the junior
class attended the dance. There were
at least 200 juniors at the dance. How
many students are in the junior class?
A.
B.
C.
D.
j ≤ 300
j ≥ 300
j ≥ 200
j ≤ 200
3 2
3
j  200
2 3
2
Example 6
3
• Solve  d  6.
5
5 3
5
  d6 
3 5
3
d ≤ -10
{d|d ≤ -10}
Check Your Progress
• Choose the best answer for the
following.
– Solve -1/3 x > 10.
A.
B.
C.
D.
x > 10/3
x > -10/3
x < -30
x > -30
3 1
3
  x  10 
1 3
1
Solve Inequalities by
Division
• Division Property of Inequalities:
– If you divide each side of an inequality by a positive
number, then the inequality remains true.
– For any real numbers a and b and any positive real
number c, if a > b, then a/c > b/c. And, if a < b, then
a / < b/ .
c
c
– If you divide each side of an inequality by a
negative number, the inequality symbol changes
direction.
– For any real numbers a and b and any negative real
number c, if a > b, then a/c < b/c. And, if a < b, then
a / > b/ .
c
c
– This property also holds for inequalities ≤ and ≥.
Example 7
• Solve each inequality.
a. 12k ≥ 60
12 12
k≥5
{k|k ≥ 5}
b. -8q < 136
-8
-8
q > -17
{q|q > -17}
Check Your Progress
• Choose the best answer for the
following.
A. Solve 15p < 60.
A.
B.
C.
D.
{p|p < 4}
{p|p < 45}
{p|p < 75}
{p|p > 4}
15p < 60
15 15
Check Your Progress
• Choose the best answer for the
following.
B. Solve -4z > 64.
A.
B.
C.
D.
{z|z < 16}
{z|z < -16}
{z|z > -16}
{z|z > 16}
-4z > 64
-4
-4