Download To solve inequalities.

Solving Inequalities Using
Addition
and
Subtraction
Complete each statement with <, =, or >.
ALGEBRA 1 LESSON 10-2
(For help, go to Lessons 1-4, 1-5, and 2-1.)
1. –3 + 4
2. –3 – 6
–5 + 4
4+6
3. –3.4 + 2
–3.45 + 2
Solve each equation.
4. x – 4 = 5
5. n – 3 = –5
6. t + 4 = –5
7. k + 2 = 5
3
10-2
6
Solving Inequalities Using
Solutions Addition and Subtraction
ALGEBRA 1 LESSON 10-2
1. –3 + 4
–5 + 4
1 > –1
2. –3 – 6
4–6
–9 < –2
3. –3.4 + 2
–3.45 + 2
–1.4 > –1.45
4. x – 4 = 5
x=9
5. n – 3 = –5
n = –2
6. t + 4 = –5
t = –9
7. k + 3 = 6
2
5
5
2
5
4
1
k=6– 3=6 – 6=6
10-2
Solving Inequalities Using
Addition
and
Subtraction
Solve p – 4 < 1. Graph the solutions.
ALGEBRA 1 LESSON 10-2
p–4+4<1+4
p<5
Add 4 to each side.
Simplify.
10-2
Solving Inequalities Using
Addition
and
Subtraction
Solve 8 > d – 2. Graph and check your solution.
ALGEBRA 1 LESSON 10-2
8+2>d–2+2
10 > d, or d < 10
Check: 8 = d – 2
8 10 – 2
8=8
8>d–2
8>9–2
8> 7
Add 2 to each side.
Simplify.
Check the computation.
Substitute 10 for d.
Check the direction of the inequality.
Substitute 9 for d.
10-2
Solving Inequalities Using
Addition
and
Subtraction
Solve c + 4 > 7. Graph the solutions.
ALGEBRA 1 LESSON 10-2
c+4–4>7–4
c>3
Subtract 4 from each side.
Simplify.
10-2
Solving Inequalities Using
Addition
and
Subtraction
In order to receive a B in your literature class, you must earn
ALGEBRA 1 LESSON 10-2
more than 350 points of reading credits. Last week you earned 120
points. This week you earned 90 points. How many more points must
you earn to receive a B?
Relate:
points
earned
plus
points
needed
is more
than
points
required
Define: Let p = the number of points needed.
Write:
120 + 90+
p
>
10-2
350
Solving Inequalities Using
Addition and Subtraction
ALGEBRA 1 LESSON 10-2
(continued)
120 + 90 + p > 350
210 + p > 350
210 + p – 210 > 350 – 210
p > 140
Combine like terms.
Subtract 210 from each side.
Simplify.
You must earn 141 more points.
10-2
Solving Inequalities Using
Addition
and
Subtraction
Solve each inequality. Graph the solutions.
ALGEBRA 1 LESSON 10-2
1. p – 7 > –5
2. w – 3 < –9
p>2
w < –6
3. x + 6 > 4
4. 13 > 9 + h
x > –2
4 > h, or h < 4
10-2
Solving Inequalities Using
Multiplication and Division
ALGEBRA 1 LESSON 10-2
(For help, go to Lessons 2-1 and 3-1.)
Solve each equation.
x
= –1
6
1. 8 = 1 t
2. 14 = –21x
3.
4. 5d = 32
5. 2 x = –12
6. 0.5n = 9
2
3
Write an inequality for each graph.
7.
8.
10-2
Solving Inequalities Using
Solutions Multiplication and Division
ALGEBRA 1 LESSON 10-2
1
1. 8 = 2 t
2. 14 = –21x
1
t=8
2
–21x = 14
x = – 14 = – 2
t = 8(2) = 16
3. x = –1
6
21
4. 5d = 32
d = 32 = 6.4
x = –1(6) = –6
5.
2
x = –12
3
3 • 2 x = 3 • –12
2
2 3
5
6. 0.5n = 9
0.5n
= 9
0.5
0.5
x = –18
7. x < –1
n = 18
8. x > 3
10-2
3
Solving Inequalities Using
Multiplication
and
Division
Solve > –2. Graph and check the solutions.
ALGEBRA 1 LESSON 10-2
z
3
3
z
3
( ) > 3(–2)
z > –6
Check:
z
= –2
3
6
– 3 = –2
–2 = –2
z
> –2
3
3
– 3 > –2
–1 > –2
Multiply each side by 3. Do not reverse the
inequality symbol.
Simplify each side.
Check the computation.
Substitute –6 for z.
Simplify.
Check the direction of the inequality.
Substitute –3 for z.
Simplify.
10-2
Solving Inequalities Using
Multiplication
and
Division
Solve 3 < – x. Graph and check the solutions.
ALGEBRA 1 LESSON 10-2
3
5
5
–3
( )
5
(3) > – 3
3
x
5
( )( )
Multiply each side by the reciprocal of – 3 , which
5
5
is – , and reverse the inequality symbol.
3
–5 > x, or x < –5
Simplify.
10-2
Solving Inequalities Using
Multiplication and Division
ALGEBRA 1 LESSON 10-2
(continued)
Check: 3 = – 3 x
5
3 = – 3 (–5)
5
Check the computation.
Substitute –5 for x.
3=3
3 < –3 x
Check the direction of the inequality.
3 < – 3 (–10)
Substitute –10 for x.
5
5
3< 6
10-2
Solving Inequalities Using
Multiplication
and
Division
Solve –4c < 24. Graph the solutions.
ALGEBRA 1 LESSON 10-2
–4c
24
>
–4
–4
Divide each side by –4. Reverse the inequality symbol.
c > –6
Simplify.
10-2
Solving Inequalities Using
Multiplication
and
Division
Your family budgets $160 to spend on fuel for a trip. How
ALGEBRA 1 LESSON 10-2
many times can they fill the car’s gas tank if it cost $25 each time?
Relate:
cost per
times
tank
number
of tanks
is at
most
total fuel
budget
Define: Let t = the number of tanks of gas.
Write:
25
•
t
<
25t < 160
25t 160
< 25
25
t < 6.4
Divide each side by 25.
Simplify.
Your family can fill the car’s tank at most 6 times.
10-2
160
Solving Inequalities Using
Multiplication
and Division
Solve
each inequality. Graph the solution.
ALGEBRA 1 LESSON 10-2
y
1. 2 > –3
y > –6
2. – p < –1
3
p>3
3. 6x < 30
x<5
4. 48 > –12h
–4 < h, or h > –4
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
(For help, go to Lessons 2-2 and 2-3.)
Solve each equation, if possible. If the equation is an identity or if it has no
solution, write identity or no solution.
1. 3(c + 4) = 6
2. 3t + 6 = 3(t – 2)
3. 5p + 9 = 2p – 1
4. 7n + 4 – 5n = 2(n + 2)
5. 1 k – 2 + k = 7
6. 2t – 32 = 5t + 1
2
3
6
Find the missing dimension of each rectangle.
7.
8.
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solutions
2. 3t + 6 = 3(t – 2)
1. 3(c + 4) = 6
3t + 6 = 3t – 6
c+4=2
c = –2
6 = –6
no solution
3. 5p + 9 = 2p – 1
4. 7n + 4 – 5n = 2(n + 2)
3p = –10
2n + 4 = 2n + 4
p = –3 1
identity
3
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solutions
5. 1 k – 2 + k = 7
2
7.
3
6
3 k = 11
6
2
11
k=
= 12
6
9
P = 2( + w)
6. 2t – 32 = 5t + 1
–3t = 33
t = –11
8.
110 = 2( + 15)
55 =
P = 2( + w)
78 = 2(26 + w)
+ 15
39 = 26 + w
40 =
13 = w
length = 40 cm
width = 13 in.
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solve 5 + 4b < 21.
5 + 4b – 5 < 21 – 5
4b < 16
Simplify.
4b 16
< 4
4
Divide each side by 4.
b<4
Check:
Subtract 5 from each side.
5 + 4b = 21
5 + 4(4)
21
Simplify.
Check the computation.
Substitute 4 for b.
21 = 21
5 + 4b < 21
5 + 4(3) < 21
Check the direction of the inequality.
Substitute 3 for b.
17 < 21
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
The band is making a rectangular banner that is 20 feet long
with trim around the edges. What are the possible widths the banner
can be if there is no more than 48 feet of trim?
Relate:
twice the
length
Write:
2(20)
plus
twice the
width
can be no
more than
+
2w
<
10-2
the length
of trim
48
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
(continued)
2(20) + 2w < 48
40 + 2w < 48
40 + 2w – 40 < 48 – 40
Simplify 2(20).
Subtract 40 from each side.
2w < 8
Simplify.
2w < 8
2
2
Divide each side by 2.
w< 4
Simplify.
The banner’s width must be 4 feet or less.
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solve 3x + 4(6 – x) < 2.
3x + 24 – 4x < 2
–x + 24 < 2
–x + 24 – 24 < 2 – 24
Use the Distributive Property.
Combine like terms.
Subtract 24 from each side.
–x < –22
Simplify.
–x –22
>
–1
–1
Divide each side by –1. Reverse the
inequality symbol.
x > 22
Simplify.
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solve 8z – 6 < 3z + 12.
8z – 6 – 3z < 3z + 12 – 3z
Subtract 3z from each side.
5z – 6 < 12
Combine like terms.
5z – 6 + 6 < 12 + 6
Add 6 to each side.
5z < 18
Simplify.
5z
18
<
5
5
Divide each side by 5.
3
z < 35
Simplify.
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solve 5(–3 + d) < 3(3d – 2).
–15 + 5d < 9d – 6
–15 + 5d – 9d < 9d – 6 – 9d
Use the Distributive Property.
Subtract 9d from each side.
–15 – 4d < –6
Combine like terms.
–15 – 4d + 15 < –6 + 15
Add 15 to each side.
–4d < 9
Simplify.
–4d
9
>
–4
–4
Divide each side by –4. Reverse the
inequality symbol.
1
d > –2 4
Simplify.
10-2
ALGEBRA 1 LESSON 10-2
Solving Multi-Step Inequalities
Solve each inequality.
1. 8 + 5a > 23
2. – 1 p < 1 p – 6
3
2
1
p>7
5
a>3
3. 3(x – 4) > 4x + 7
4. 3(3c + 2) < 2(3c – 2)
1
c < –3 3
x < –19
10-2