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Pre-Algebra
10-4
Solving Multistep Inequalities
Solve.
1. 6x + 36 = 2x
2. 4x – 13 = 15 + 5x
3. 5(x – 3) = 2x + 3
4. + x =
x = –9
x = –28
x = 678
316
1116
x = –
Warm Up
Learn to solve two-step inequalities and graph the solutions of an inequality on a number line.
Solving a multistep inequality uses the same inverse operations as solving a multistep equation. Multiplying or dividing the inequality by a negative number reverses the inequality symbol.
Solve and graph.
A. 4x + 1 > 13
4x + 1 > 13 – 1 – 1 Subtract 1 from both sides.
4x > 124x4
> 124
Divide both sides by 4.
x > 3 1 2 3 4 5 6 7
Example
B. –7 < 3x + 8
–7 < 3x + 8
– 8 – 8 Subtract 8 from both sides.
–15 < 3x
– 15 3
< 3x 3 Divide both sides by 3.
–5 < x -7 -6 -5 -4 -3 -2 -1
Example
C. -9x + 7 25
–9x + 7 25
– 7 – 7 Subtract 7 from both sides.
–9x 18
–9x–9
18–9
Divide each side by –9; change to .
x –2-6 -5 -4 -3 -2 -1 0
Example
Solve and graph.
A. 5x + 2 > 12
5x + 2 > 12 – 2 – 2 Subtract 2 from both sides.
5x > 105x5
> 105
Divide both sides by 5.
x > 2 1 2 3 4 5 6 7
Try This
B. –5 < 2x + 9
–5 < 2x + 9
– 9 – 9 Subtract 9 from both sides.
–14 < 2x
– 14 2
< 2x 2 Divide both sides by 2.
–7 < x -7 -6 -5 -4 -3 -2 -1
Try This
C. -4x + 2 18
–4x + 2 18
– 2 – 2 Subtract 2 from both sides.
–4x 16
–4x–4
16–4
Divide each side by –4; change to .
x –4-6 -5 -4 -3 -2 -1 0
Try This
Solve and graph.
A. 10x + 21 – 4x < –15
10x + 21 – 4x < –15
– 21 – 21 Subtract 21 from both sides.
6x6
< –36 6 Divide both sides by 6.
x < –6-8 -7 -6 -5 -4 -3 -2
6x + 21 < –15 Combine like terms.
6x < –36
Example
Multiply by LCD, 20.
8x + 15 18
– 15 – 15 Subtract 15 from both sides.
8x 3
B. + 2x5
34
910
+ 2x5
34
910
20( + ) 20( )2x5
34
910
20( ) + 20( ) 20( )2x5
34
910
Example
x 38
8x8
38 Divide both sides by 8.
8x 3
0 1
38
Example Continued
C. 8x + 8 > 11x – 1
8x + 8 > 11x – 1– 8x – 8x Subtract 8x from both sides.
8 > 3x – 1
93
> 3x 3
Add 1 to each side.
3 > x
-1 0 1 2 3 4 5
+1 +1
9 > 3x
Divide both sides by 3.
Example
Solve and graph.
A. 15x + 30 – 5x < –10
15x + 30 – 5x < –10
– 30 – 30 Subtract 30 from both sides.
10x10
< –40 10 Divide both sides by 10.
x < –4-8 -7 -6 -5 -4 -3 -2
10x + 30 < –10 Combine like terms.
10x < –40
Try This
Multiply by LCD, 20.
12x + 5 10
– 5 – 5 Subtract 5 from both sides.
12x 5
B. + 3x5
14
510
+ 3x5
14
510
20( + ) 20( )3x5
14
510
20( ) + 20( ) 20 ( )3x5
14
510
Try This
x 512
12x12
512 Divide both sides by 12.
12x 5
0 5 12
Try This Continued
C. 4x + 3 > 8x – 1
4x + 3 > 8x – 1– 4x – 4x Subtract 4x from both sides.
3 > 4x – 1
44
> 4x 4
Add 1 to each side.
1 > x
-1 0 1 2 3 4 5
+1 +1
4 > 4x
Divide both sides by 4.
Try This
A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit?Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost.
R > C
Example: Business Application
The revenue from selling x bumper stickers at $1.25 each is 1.25x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or 55 + 0.15x. Substitute the expressions for R and C.
1.25x > 55 + 0.15x Let x represent the number of bumper stickers sold. Fixed cost is $55. Unit cost is 15 cents.
Example Continued
– 0.15x – 0.15x Subtract 0.15x from both sides.
1.10x > 55
x > 50
The Spanish club must sell more than 50 bumper stickers to make a profit.
Divide both sides by 1.10.
1.25x > 55 + 0.15x
1.10x1.10
551.10>
Example Continued
R > C
A school’s Spanish club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit?Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost.
Try This
The revenue from selling x bumper stickers at $2.50 each is 2.5x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or 45 + 0.25x. Substitute the expressions for R and C.
2.5x > 45 + 0.25x Let x represent the number of bumper stickers sold. Fixed cost is $45. Unit cost is 25 cents.
Try This Continued
– 0.25x – 0.25x Subtract 0.25x from both sides.
2.25x > 45
x > 20
The Spanish club must sell more than 20 bumper stickers to make a profit.
Divide both sides by 2.25.
2.5x > 45 + 0.25x
2.25x2.25
452.25>
Try This Continued
Solve and graph.
1. 4x – 6 > 10
2. 7x + 9 < 3x – 15
3. w – 3w < 32
4. w +
x < –6
x > 4
w > –1623
14
12
w 38
1 2 3 4 5 6 7
-10 -9 -8 -7 -6 -5 -4
-18 -17 -16 -15 -14 -13 -12
0 38
Lesson Quiz: Part 1
5. Antonio has budgeted an average of $45 a month for entertainment. For the first five months of the year he has spent $48, $39, $60, $48, and $33. How much 1 can Antonio spend in the sixth month without exceeding his average budget?
no more than $42
Lesson Quiz: Part 2
