Transcript la1_ch05_03 Solve Multi-Step Inequalities_teacher

5.3 Solve Multi-Step Inequalities
• You will solve multistep inequalities.
• Essential question:
• How do you solve
multi-step inequalities?
You will learn how to answer this
question by using the distributive
property and the properties of
inequality.
EXAMPLE
Warm-Up1Exercises
Solve a two-step inequality
Solve 3x – 7 < 8. Graph your solution.
3x – 7 < 8
3x < 15
x<5
Write original inequality.
Add 7 to each side.
Divide each side by 3.
ANSWER
The solutions are all real numbers less than 5. Check
by substituting a number less than 5 in the original
inequality.
EXAMPLE
Warm-Up1Exercises
Solve a two-step inequality
CHECK
3x –7 < 8
?
3(0) – 7 < 8
–7 < 8
Write original inequality.
Substitute 0 for x.
Solution checks.
EXAMPLE
Warm-Up2Exercises
Solve a multi-step inequality
Solve –0.6(x – 5)  15.
–0.6(x – 5)  15
–0.6x + 3  15
–0.6x  12
x  –20
Write original inequality.
Distributive property
Subtract 3 from each side.
Divide each side by – 0.6. Reverse
inequality symbol.
Is -21 a solution of the inequality? Explain.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the inequality. Graph your solution.
1.
2x – 5 <
– 23.
ANSWER
x<
– 14
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the inequality. Graph your solution.
2.
–6y + 5 <
– –16.
ANSWER
y>
– 3.5
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Solve the inequality. Graph your solution.
3.
–1 ( p –12) > –2
4
ANSWER
p < 20
EXAMPLE
Warm-Up3Exercises
Solve a multi-step inequality
Solve 6x – 7 > 2x + 17. Graph your solution.
6x – 7 > 2x + 17
Write original inequality.
6x > 2x + 24
Add 7 to each side.
4x > 24
Subtract 2x from each side.
x>6
Divide each side by 4.
ANSWER
The solutions are all real numbers greater than 6.
EXAMPLE
Warm-Up4Exercises
Identify the number of solutions of an inequality
Solve the inequality, if possible.
a.
14x + 5 < 7(2x – 3)
SOLUTION
a.
14x + 5 < 7(2x – 3)
Write original inequality.
14x + 5 < 14x – 21
Distributive property
5 < –21
Subtract 14x from each side.
ANSWER
There are no solutions because 5 < –21 is false.
EXAMPLE
Warm-Up4Exercises
Identify the number of solutions of an inequality
b.
12x – 1 > 6(2x – 1)
12x – 1 > 6(2x – 1)
Write original inequality.
12x – 1 > 12x – 6
Distributive property
–1 > –6
Subtract 12x from each side.
ANSWER
All real numbers are solutions because –1 > –6 is true.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Solve the inequality, if possible. Graph your solution.
4.
5x – 12  3x – 4
ANSWER
x4
EXAMPLE
4Exercises
Warm-Up
fornumber
Examples
3 and 4 of an inequality
Identify the
of solutions
GUIDED
PRACTICE
Solve the inequality, if possible. Graph your solution.
5.
5(m + 5) < 5m + 17
ANSWER
no solution
EXAMPLE
4Exercises
Warm-Up
fornumber
Examples
3 and 4 of an inequality
Identify the
of solutions
GUIDED
PRACTICE
Solve the inequality, if possible. Graph your solution.
6.
1 – 8s <
– –4(2s – 1)
ANSWER
all real numbers
Warm-Up5Exercises
EXAMPLE
Solve a multi-step problem
Car Wash
Use the sign shown. A gas station
charges $.10 less per gallon of
gasoline if a customer also gets a
car wash. What are the possible
amounts (in gallons) of gasoline
that you can buy if you also get a
car wash and can spend at most
$20?
Warm-Up5Exercises
EXAMPLE
Solve a multi-step problem
SOLUTION
Because you are getting a car wash, you will pay
$2.09 – $.10 = $1.99 per gallon of gasoline. Let g be
the amount (in gallons) of gasoline that you buy.
STEP 1
Write a verbal model. Then write an inequality.
Price of
gasoline
(dollars/gallon)
1.99
•
Amount of
gasoline
(gallons)
g
Price of
car wash
(dollars)
+
+
8
<
Maximum
amount
(dollars)
<
–
20
Warm-Up5Exercises
EXAMPLE
Solve a multi-step problem
STEP 2
Solve the inequality.
1.99g + 8 ≤ 20
1.99g ≤ 12
g ≤ 6.03015. . .
Write inequality.
Subtract 8 from each side.
Divide each side by 1.99.
ANSWER
You can buy up to slightly more than 6 gallons of
gasoline.
Warm-Up5Exercises
EXAMPLE
Solve a multi-step problem
CHECK
You can use a table to
check the reasonableness
of your answer. The table
shows that you will pay
$19.94 for exactly 6
gallons of gasoline.
Because $19.94 is less
than $20, it is reasonable
to conclude that you can
buy slightly more than 6
gallons of gasoline.
Gasoline Total amount
spent
(goal)
(dollars)
0
8.00
1
9.99
2
3
4
11.98
13.97
15.96
5
17.95
6
19.94
Warm-Up
Exercises
GUIDED
PRACTICE
GUIDED PRACTICE
for
for Example
Example 53 and 4
7. WHAT IF? In Example 5, suppose that a car wash
costs $9 and gasoline regularly costs $2.19 per gallon.
What are the possible amounts (in gallons) of
gasoline that you can buy?
ANSWER
g ≤ 5.02 gal
Warm-Up
EXAMPLE
5Exercises
for Example
5
Solve a multi-step
problem
GUIDED
PRACTICE
Camp Costs
8. You are saving money for a summer camp that
costs $1800. You have saved $500 so far, and you
have 14 more weeks to save the total amount.
What are the possible average amounts of
money that you can save per week in order to
have a total of at least $1800 saved?
ANSWER
at least $92.86/wk
• You will solve multistep inequalities.
• You can solve multi-step
inequalities by applying the
same steps used for solving
multi-step equations, with the
additional step of reversing
an inequality symbol if you
multiply or divide by a
negative number.
• Essential question:
• How do you solve
multi-step inequalities?
Use the distributive property and
the properties of inequality to
produce equivalent inequalities.
Reverse the direction of the
inequality symbol when dividing
or multiplying by a negative
number.
If you reach an inequality that
is false, there are no solutions; if
you reach an inequality that is
always true, then all real
numbers are solutions.
Daily
Homework
Quiz
Warm-Up
Exercises
1. Solve 3(x + 8) < 9. Graph your solution.
ANSWER
all real numbers less than –5
Daily
Homework
Quiz
Warm-Up
Exercises
Solve the inequality, if possible.
2.
6k + 1 > 3(2k – 3)
ANSWER
all real numbers because 1 > –9 is true
3. 5(a – 2)< 5a – 14
ANSWER
no solutions because –10 < –14 is false
Daily
Homework
Quiz
Warm-Up
Exercises
4. A box of cat treats contains at least 50 treats. So
far you have fed your cats 18 treats. If you want the
box of treats to last 8 days, what are the possible
average numbers of treats you can feed the cats per
day?
ANSWER
an average of no more than 4 treats per day.