Transcript Solve the inequality.

Solving Inequalities with
3-5 Variables on Both Sides
Warm Up
Lesson Presentation
Lesson Quiz
Solving Inequalities with
3-5 Variables on Both Sides
Warm Up
Solve each equation.
1. 2x = 7x + 15 x = –3
2. 3y – 21 = 4 – 2y
y=5
3. 2(3z + 1) = –2(z + 3) z = –1
4. 3(p – 1) = 3p + 2
no solution
5. Solve and graph 5(2 – b) > 52. b < –3
–6
–5
–4
–3
–2
–1
0
Solving Inequalities with
3-5 Variables on Both Sides
Sunshine State Standards
MA.912.A.3.5 Symbolically represent and
solve multi-step and real-world applications
that involve linear…inequalities.
Also MA.912.A.3.4, MA.912.A.10.3.
Solving Inequalities with
3-5 Variables on Both Sides
Objective
Solve inequalities that contain variable
terms on both sides.
Solving Inequalities with
3-5 Variables on Both Sides
Some inequalities have variable terms on both
sides of the inequality symbol. You can solve
these inequalities like you solved equations with
variables on both sides.
Use the properties of inequality to “collect” all the
variable terms on one side and all the constant
terms on the other side.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 1A: Solving Inequalities with
Variables on Both Sides
Solve the inequality and graph the solutions.
y ≤ 4y + 18
To collect the variable terms on one
y ≤ 4y + 18
side, subtract y from both sides.
–y –y
0 ≤ 3y + 18
–18
– 18
Since 18 is added to 3y, subtract 18
from both sides to undo the
addition.
–18 ≤ 3y
Since y is multiplied by 3, divide both
sides by 3 to undo the
multiplication.
–6 ≤ y (or y  –6)
–10 –8 –6 –4 –2
0
2
4
6
8 10
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 1B: Solving Inequalities with
Variables on Both Sides
Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
To collect the variable terms on one
–2m
– 2m
side, subtract 2m from both sides.
2m – 3 <
+6
+3
+3
2m
<
9
Since 3 is subtracted from 2m, add
3 to both sides to undo the
subtraction
Since m is multiplied by 2, divide
both sides by 2 to undo the
multiplication.
4
5
6
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 1a
Solve the inequality and graph the solutions.
4x ≥ 7x + 6
4x ≥ 7x + 6
–7x –7x
To collect the variable terms on one
side, subtract 7x from both sides.
–3x ≥ 6
x ≤ –2
Since x is multiplied by –3, divide
both sides by –3 to undo the
multiplication. Change ≥ to ≤.
–10 –8 –6 –4 –2
0
2
4
6
8 10
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 1b
Solve the inequality and graph the solutions.
–3
–3
To collect the variable terms on
one side, subtract 3 from both
sides.
Subtract one-fourth t from both
sides.
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 1b Continued
Solve the inequality and graph the solutions.
Divide both sides by ten-fourths.
–5 –4 –3 –2 –1
0
1
2
3
4
5
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 2: Business Application
The Home Cleaning Company charges $312 to
power-wash the siding of a house plus $12 for
each window. Power Clean charges $156, to
power-washing the siding plus $24 per window.
How many windows must a house have to make
the total cost from The Home Cleaning
Company less expensive than Power Clean?
Let w be the number of windows.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 2 Continued
Home
Cleaning
Company
siding
charge
312
plus
+
$12 per
window
times
# of
windows
12
•
w
is
less
than
Power
Clean
siding
charge
plus
<
156
+
$24 per
window
24
times
# of
windows.
•
w
312 + 12w < 156 + 24w To collect like terms, subtract 12w
and 156 from both sides.
−156 – 12w −156 –12w
156
<
12w
Since w is multiplied by 12, divide
both sides by 12 to undo the
multiplication.
13 < w
The Home Cleaning Company is less expensive for
houses with more than 13 windows.
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 2
A-Plus Advertising charges a fee of $24 plus
$0.10 per flyer to print and deliver flyers. Print
and More charges $0.25 per flyer. For how
many flyers is the cost at A-Plus Advertising
less than the cost of Print and More?
Let f represent the number of flyers printed.
A-Plus
Advertising plus
fee of $24
24
+
$0.10
per
flyer
times
0.10
•
Print and
# of
flyers
is less
than
More’s cost
f
<
0.25
times
# of
flyers.
per flyer
•
f
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 2 Continued
24 + 0.10f < 0.25f
–0.10f –0.10f
24
To collect the variable terms,
subtract 0.10f from both sides.
< 0.15f
Since f is multiplied by 0.15,
divide both sides by 0.15 to
undo the multiplication.
160 < f
More than 160 flyers must be delivered to make
A-Plus Advertising the lower cost company.
Solving Inequalities with
3-5 Variables on Both Sides
You may need to simplify one or both sides of
an inequality before solving it. Look for like
terms to combine and places to use the
Distributive Property.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 3A: Simplify Each Side Before
Solving
Solve the inequality and graph the solutions.
2(k – 3) > 6 + 3k – 3
Distribute 2 on the left side of
2(k – 3) > 3 + 3k
the inequality.
2k + 2(–3) > 3 + 3k
2k – 6 > 3 + 3k
–2k
– 2k
–6 > 3 + k
–3 –3
–9 > k
To collect the variable terms,
subtract 2k from both
sides.
Since 3 is added to k, subtract 3
from both sides to undo the
addition.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 3A Continued
Solve the inequality and graph the solutions.
–9 > k
–12
–9
–6
–3
0
3
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 3B: Simplify Each Side Before
Solving
Solve the inequality and graph the solution.
3.2y - 2.3y ≥ 0.4 y - 0.5
3.2y − 2.3y ≥ 0.4y – 0.5
Combine y terms.
0.9y ≥ 0.4y – 0.5
–0.4y –0.4y
To collect the variable terms,
subtract 0.4y from both
0.5y ≥ – 0.5
sides.
0.5y ≥ –0.5
0.5
0.5
y ≥ –1
–5 –4 –3 –2 –1
0
1
2
3
Since y is multiplied by 0.5,
divide both sides by 0.5 to
undo the multiplication.
4
5
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 3a
Solve the inequality and graph the solutions.
5(2 – r) ≥ 3(r – 2)
Distribute 5 on the left side of the
inequality and distribute 3 on
5(2 – r) ≥ 3(r – 2)
the right side of the inequality.
5(2) – 5(r) ≥ 3(r) + 3(–2)
Since 6 is subtracted from 3r,
10 – 5r ≥ 3r – 6
add 6 to both sides to undo
+6
+6
the subtraction.
16 − 5r ≥ 3r
Since 5r is subtracted from 16
+ 5r +5r
add 5r to both sides to undo
the subtraction.
16
≥ 8r
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 3a Continued
Solve the inequality and graph the solutions.
16 ≥ 8r
Since r is multiplied by 8, divide
both sides by 8 to undo the
multiplication.
2≥r
–6
–4
–2
0
2
4
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 3b
Solve the inequality and graph the solutions.
0.5x – 0.3 + 1.9x < 0.3x + 6
2.4x – 0.3 < 0.3x + 6
2.4x – 0.3 < 0.3x + 6
+ 0.3
+ 0.3
2.4x
–0.3x
2.1x
< 0.3x + 6.3
–0.3x
<
6.3
Simplify.
Since 0.3 is subtracted
from 2.4x, add 0.3 to
both sides.
Since 0.3x is added to
6.3, subtract 0.3x from
both sides.
Since x is multiplied by
2.1, divide both sides
by 2.1.
x<3
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 3b Continued
Solve the inequality and graph the solutions.
x<3
–5 –4 –3 –2 –1
0
1
2
3
4
5
Solving Inequalities with
3-5 Variables on Both Sides
Some inequalities are true no matter what value is
substituted for the variable. For these inequalities,
all real numbers are solutions.
Some inequalities are false no matter what value
is substituted for the variable. These inequalities
have no solutions.
If both sides of an inequality are fully simplified
and the same variable term appears on both sides,
then the inequality has all real numbers as
solutions or it has no solutions. Look at the other
terms in the inequality to decide which is the case.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 4A: All Real Numbers as
Solutions or No Solutions
Solve the inequality.
2x – 7 ≤ 5 + 2x
2x – 7 ≤ 5 + 2x
The same variable term (x) appears on both
sides. Look at the other terms.
For any number x, subtracting 7 will always
result in a lesser number than adding 5.
All values of x make the inequality true.
All real numbers are solutions.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 4B: All Real Numbers as
Solutions or No Solutions
Solve the inequality.
2 (3y – 2) – 4 ≥ 3(2y + 7)
Distribute 2 on the left side
2(3y−2) − 4 ≥ 3(2y +7) and 3 on the right side of
the inequality. Add -4’s on
the left side.
6y − 4 − 4 ≥ 6y + 21
6y − 8 ≥ 6y + 21
6y − 8 ≥ 6y + 21
The same variable term (y) appears on both
sides. Look at the other terms.
Solving Inequalities with
3-5 Variables on Both Sides
Additional Example 4B Continued
Solve the inequality.
2 (3y – 2) – 4 ≥ 3(2y + 7)
6y − 8 ≥ 6y + 21
For any number y, subtracting 8 will always
result in a lesser number than adding 21.
No values of y make the inequality true.
There are no solutions.
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 4a
Solve the inequality.
4(y – 1) ≥ 4y + 2
4(y – 1) ≥ 4y + 2
Distribute 4 on the left side.
4(y) + 4(–1) ≥ 4y + 2
4y – 4 ≥ 4y + 2
–4y
–4y
–4 ≥ 2 
Subtract 4y from both sides.
False statement.
No values of y make the inequality true.
There are no solutions.
Solving Inequalities with
3-5 Variables on Both Sides
Check It Out! Example 4b
Solve the inequality.
x–2<x+1
x–2<x+1
The same variable term (x) appears on both
sides. Look at the other terms.
For any number x, subtracting 2 will always
result in a lesser number than adding 1.
All values of x make the inequality true.
All real numbers are solutions.
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. t < 5t + 24 t > –6
2. 5x – 9 ≤ 4.1x – 81 x ≤ –80
3. 4b + 4(1 – b) > b – 9
b < 13
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz: Part II
4. Rick bought a photo printer and supplies for
$186.90, which will allow him to print photos
for $0.29 each. A photo store charges $0.55
to print each photo. How many photos must
Rick print before his total cost is less than
getting prints made at the photo store?
Rick must print more than 718 photos.
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz: Part III
Solve each inequality.
5. 2y – 2 ≥ 2(y + 7)
no solutions
6. 2(–6r – 3) < –3(4r + 2)
all real numbers
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz for Student Response Systems
1. Identify the solution for the inequality.
a < 6a + 45
A.
a > –9
B.
a<–9
C.
a>5
D.
a<5
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz for Student Response Systems
2. Identify the solution for the inequality.
6t + 4 ≤ 5.4t − 32
A.
t ≤ –6
B.
t ≥ –6
C.
t ≤ –60
D.
t ≥ –60
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz for Student Response Systems
3. Identify the solution for the inequality.
7y + 7(3 − y) > 2y − 9
A.
y < 10
B.
y < –10
C.
y < –15
D.
y < 15
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz for Student Response Systems
4. John bought a computer scanner and supplies
for $215.70, which will allow him to scan
images for $0.34 each. A computer center
charges $0.59 to scan each image. How many
images must John scan before his total cost is
less than getting scanned images at the
computer center?
A. John must scan more than 863 images.
B. John must scan more than 862 images.
C. John must scan more than 788 images.
D. John must scan more than 768 images.
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz for Student Response Systems
5. Solve the inequality.
3y − 4 ≥ 3(y + 4)
A. all real numbers
B. no solutions
Solving Inequalities with
3-5 Variables on Both Sides
Lesson Quiz for Student Response Systems
6. Solve the inequality.
–2(8x − 2) ≤ 4(–4x + 1)
A. all real numbers
B. no solutions