8.4 Solving Multi-Step Inequalities - Big Ideas Math
English
8.4
Spanish
Solving Multi-Step Inequalities
How can you use an inequality to describe the
SSTATE
STANDARDS
area and perimeter of a composite figure?
MA.8.A.4.2
1
ACTIVITY: Areas and Perimeters of Composite Figures
Work with a partner.
5
a. For what values of x will the area
of the blue region be greater than
12 square units?
x
3
1
b. For what values of x will the sum
of the inner and outer perimeters
of the blue region be greater than
20 units?
c. For what values of y will the area of
the trapezoid be less than or equal
to 10 square units?
4
3
d. For what values of y will the perimeter
of the trapezoid be less than or equal
to 16 units?
y
w
e. For what values of w will the area
of the red region be greater than or
equal to 36 square units?
f.
For what values of w will the sum
of the inner and outer perimeters
of the red region be greater than
47 units?
x
4
Chapter 8
Linear Inequalities
6
8
g. For what values of x will the area
of the yellow region be less than
4¦Ð square units?
4
334
10
h. For what values of x will the sum
of the inner and outer perimeters
of the yellow region be less than
4¦Ð + 20 units?
English
Spanish
2
ACTIVITY: Volume and Surface Area of a Composite Solid
Work with a partner.
a. For what values of x will
the volume of the solid be
greater than or equal to
42 cubic units?
4
b. For what values of x will the
surface area of the solid be
greater than 72 square units?
3
2
3
x
3
ACTIVITY: Planning a Budget
Work with a partner.
You are building a patio. You want to cover the patio with Spanish tile that
costs $5 per square foot. Your budget for the tile is $1700. How wide can you
make the patio without going over your budget?
Tiles are
needed
under the
plants.
No tile is
needed
under the
hot tub.
6 ft
6 ft
24 ft
4. IN YOUR OWN WORDS How can you use an inequality to describe the
area and perimeter of a composite figure? Give an example. Include a
diagram with your example.
Use what you learned about solving multi-step inequalities to
complete Exercises 3 and 4 on page 338.
Section 8.4
Solving Multi-Step Inequalities
335
English
Spanish
Lesson
8.4
Lesson Tutorials
You can solve multi-step inequalities the same way you solve
multi-step equations.
EXAMPLE
1
Solving Two-Step Inequalities
a. Solve 5x ? 4 ¡Ý 11. Graph the solution.
5x ? 4 ¡Ý 11
+4
Step 1: Undo the subtraction.
Write the inequality.
+4
Add 4 to each side.
5x ¡Ý 15
Simplify.
5x
5
Divide each side by 5.
15
5
¡ª ¡Ý ¡ª
Step 2: Undo the multiplication.
x¡Ý3
Simplify.
The solution is x ¡Ý 3.
x¡Ý3
?3
?2
?1
0
1
2
3
4
5
6
7
Check: x = 4 is a solution.
Check: x = 0 is not a solution.
y
?6
b. Solve ¡ª + 7 < 9. Graph the solution.
y
?6
¡ª+7 < 9
?7
Write the inequality.
?7
Subtract 7 from each side.
y
?6
¡ª < 2
? ?6y
Simplify.
?
Multiply each side by ?6. Reverse the
inequality symbol.
?6 ¡ª > ?6 2
y > ?12
Simplify.
The solution is y > ?12.
y > ?12
?18 ?16 ?14 ?12 ?10
?8
?6
?4
?2
0
2
Solve the inequality. Graph the solution.
Exercises 5¨C10
336
Chapter 8
MSFL8PE_0804.indd 336
1. 4b ? 1 < 7
2.
8 + 9c ¡Ý ?28
3.
n
?2
¡ª + 11 > 12
Linear Inequalities
10/20/09 4:31:48 PM
English
Spanish
EXAMPLE
2
Standardized Test Practice
Which graph represents the solution of ?7(x + 3) ¡Ü 28?
A
¡ð
C
¡ð
?10 ?9
4
5
B
¡ð
?8
?7
?6
?5
?4
6
7
8
9
10
?10 ?9
D
¡ð
4
5
?7(x + 3) ¡Ü
28
Write the inequality.
?7x ? 21 ¡Ü
28
Use Distributive Property.
+ 21
+ 21
?8
?7
?6
?5
?4
6
7
8
9
10
Add 21 to each side.
?7x ¡Ü 49
Simplify.
?7x
?7
Divide each side by ?7. Reverse the
inequality symbol.
49
?7
¡ª ¡Ý ¡ª
x ¡Ý ?7
Simplify.
The correct answer is ¡ð
B .
EXAMPLE
3
Real-Life Application
You need a mean score of at least 90 to advance to the next round of
the trivia game. What score do you need on the fifth game to advance?
Use the definition of mean to write and solve an inequality. Let x be the
score on the fifth game.
The phrase ¡°at least¡± means
95 + 91 + 77 + 89 + x
greater than or equal to.
¡ª¡ª ¡Ý 90
5
352 + x
5
352 + x
5 ¡ª ¡Ý 5 90
5
¡ª ¡Ý 90
?
Remember
The mean in Example
3 is equal to the sum
of the game scores
divided by the number
of games.
?
352 + x ¡Ý
? 352
450
? 352
x ¡Ý 98
Simplify.
Multiply each side by 5.
Simplify.
Subtract 352 from each side.
Simplify.
You need at least 98 points to advance to the next round.
Solve the inequality. Graph the solution.
Exercises 12¨C17
4. 2(k ? 5) < 6
5.
?4(n ? 10) < 32
6.
?3 ¡Ü 0.5(8 + y)
7. WHAT IF? In Example 3, you need a mean score of at least 88 to
advance to the next round of the trivia game. What score do you
need on the fifth game to advance?
Section 8.4
Solving Multi-Step Inequalities
337
English
Spanish
Exercises
8.4
Help with Homework
1. WRITING Compare and contrast solving multi-step inequalities and solving
multi-step equations.
2. OPEN-ENDED Describe how to solve the inequality 3(a + 5) < 9.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
3. For what values of k will the
perimeter of the octagon be
less than or equal to 64 units?
4. For what values of h will the
surface area of the solid be
greater than 46 square units?
k
4
4
1
k
2
h
1
k
2
4
3
5
4
k
Solve the inequality. Graph the solution.
1
5. 7b + 4 ¡Ý 11
4
5
11
5
8. ¡ª < 3w ? ¡ª
m
3
7. 1 ? ¡ª ¡Ü 6
6. 2v ? 4 < 8
9. 1.8 < 0.5 ? 1.3p
11. ERROR ANALYSIS Describe and correct
the error in solving the inequality.
10. ?2.4r + 9.6 ¡Ý 4.8
?
x
4
¡ª+6 ¡Ý 3
x + 6 ¡Ý 12
x¡Ý6
Solve the inequality. Graph the solution.
2 12. 6( g + 2) ¡Ü 18
1
3
15. ?¡ª(u + 2) > 5
5
3
13. 2( y ? 5) ¡Ü 16
14. ?10 ¡Ý ¡ª(h ? 3)
16. 2.7 > 0.9(n ? 1.7)
17. 10 > ?2.5(z ? 3.1)
18. ATM Write and solve an inequality that
represents the number of $20 bills you can
withdraw from the account without going
below the minimum balance.
338
Chapter 8
Linear Inequalities
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