Download 3-5 Solving Inequalities with Variables on Both Sides

3-5
Solving Inequalities with Variables on Both Sides
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Warm Up
California Standards
Lesson Presentation
3-5
Solving Inequalities with 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
3-5
Solving Inequalities with Variables on Both Sides
California
Standards
4.0 Students simplify expressions before
solving linear equations and inequalities in one
variable, such as 3(2x – 5) + 4(x – 2) = 12.
5.0 Students solve multi-step problems,
including word problems, involving linear
equations and linear inequalities in one variable
and provide justification for each step.
3-5
Solving Inequalities with 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.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 1A: Solving Inequalities with
Variables on Both Sides
Solve the inequality and graph the solutions.
y ≤ 4y + 18
y ≤ 4y + 18
–y –y
0 ≤ 3y + 18
–18
–18 ≤ 3y
– 18
To collect the variable terms on one
side, subtract y from both sides.
Since 18 is added to 3y, subtract 18
from both sides to undo the
addition.
Since y is multiplied by 3, divide both
sides by 3 to undo the
multiplication.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 1A: Continued
Solve the inequality and graph the solutions.
y ≤ 4y + 18
The solution set is {y:y ≥ –6}.
–6 ≤ y (or y  –6)
–10 –8 –6 –4 –2
0
2
4
6
8 10
3-5
Solving Inequalities with Variables on Both Sides
Helpful Hint
Your first step can also be to subtract 4y from
both sides to get –3y ≤ 18. When you divide by a
negative number, remember to reverse the
inequality symbol.
3-5
Solving Inequalities with 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
–2m
– 2m
2m – 3 <
+3
2m
<
+6
+3
9
To collect the variable terms on one
side, subtract 2m from both sides.
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.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 1B Continued
Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
The solution set is {m:m
4
5
6
}.
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 1a
Solve the inequality and graph the solutions.
Check your answer.
4x ≥ 7x + 6
4x ≥ 7x + 6
–7x –7x
To collect the variable terms on one
side, subtract 7x from both sides.
–3x ≥ 6
Since x is multiplied by –3, divide
both sides by –3 to undo the
multiplication. Change ≥ to ≤.
The solution set is {x:x ≤ –2}.
x ≤ –2
–10 –8 –6 –4 –2
0
2
4
6
8 10
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 1a Continued
Solve the inequality and graph the solutions.
Check your answer.
4x ≥ 7x + 6
Check
Check the endpoint, –2.
4x =
4(–2)
–8
–8
7x + 6
7(–2) + 6
–14 + 6
–8 
Check a number less
than –2.
4x ≥ 7x + 6
4(–3) ≥ 7(–3) + 6
–12 ≥ –21 + 6
–12 ≥ –15 
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 1b
Solve the inequality
Check your answer.
5t + 1 < –2t – 6
5t + 1 < –2t – 6
+2t
+2t
7t + 1 < –6
– 1 < –1
7t
< –7
7t < –7
7
7
t < –1
–5 –4 –3 –2 –1
0
1
2
3
and graph the solutions.
To collect the variable terms on
one side, add 2t to both sides.
Since 1 is added to 7t, subtract 1
from both sides to undo the
addition.
Since t is multiplied by 7, divide
both sides by 7 to undo the
multiplication.
The solution set is {t:t < –1}.
4
5
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 1b Continued
Solve the inequality and graph the solutions.
Check your answer.
5t + 1 < –2t – 6
Check
Check the endpoint, –1.
5t + 1 = –2t – 6
5(–1) + 1 –2(–1) – 6
–5 + 1
2–6
–4
–4 
Check a number less
than –1.
5t + 1 < –2t – 6
5(–2) +1 < –2(–2) – 6
–9 < 4 – 6
–9 < –2
3-5
Solving Inequalities with 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 $36 per
window, and the price includes power-washing
the siding. 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.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 2 Continued
Home
Cleaning
Company
siding
charge
312
plus
+
$12 per
window
12
times
•
312 + 12w < 36w
– 12w –12w
312 < 24w
13 < w
# of
window
s
w
is
less
than
<
Power
Clean
cost per
window
36
times
•
# of
windows.
w
To collect the variable terms,
subtract 12w from both sides.
Since w is multiplied by 24, divide
both sides by 24 to undo the
multiplication.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 2 Continued
The Home Cleaning Company is less expensive for
houses with more than 13 windows.
3-5
Solving Inequalities with 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
•
# of
flyers
f
is less
than
<
Print and
More’s cost
times
# of
flyers.
per flyer
0.25
•
f
3-5
Solving Inequalities with 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.
3-5
Solving Inequalities with 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.
3-5
Solving Inequalities with 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.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 3A Continued
Solve the inequality and graph the solutions.
2(k – 3) > 6 + 3k – 3
The solution set is {k:k < –9}.
–9 > k
–12
–9
–6
–3
0
3
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 3B: Simplify Each Side Before
Solving
Solve the inequality and graph the solution.
0.9y ≥ 0.4y – 0.5
0.9y ≥ 0.4y – 0.5
– 0.4y – 0.4y
0.5y ≥
– 0.5
To collect the variable terms,
subtract 0.4y from both sides.
Since y is multiplied by 0.5,
divide both sides by 0.5 to
undo the multiplication.
The solution set is {y:y ≥ –1}.
0.5y ≥ –
0.5
0.5
0.5
y ≥ –1
–5 –4 –3 –2 –1
0
1
2
3
4
5
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 3a
Solve the inequality and graph the solutions.
Check your answer.
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
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 3a Continued
Solve the inequality and graph the solutions.
Check your answer.
16 ≥ 8r
Since r is multiplied by 8, divide
both sides by 8 to undo the
multiplication.
The solution set is {r:r ≤ 2}.
2≥r
–6
–4
–2
0
2
4
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 3a Continued
Solve the inequality and graph the solutions.
Check your answer.
Check
Check the endpoint, 2.
5(2 – r) = 3(r – 2)
5(2 – 2)
5(0)
0
3(2 – 2)
3(0)
0
Check a number less
than 2.
5(2 – r) ≥ 3(r – 2)
5(2 – 0) ≥ 3(0 – 2)
5(2) ≥ 3(–2)
10 ≥ –6
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 3b
Solve the inequality and graph the solutions.
Check your answer.
0.5x – 0.3 + 1.9x < 0.3x + 6
0.5x – 0.3 + 1.9x < 0.3x + 6
Combine like terms.
Since 0.3 is subtracted
2.4x – 0.3 < 0.3x + 6
from 2.4x, add 0.3 to
+ 0.3
+ 0.3
both sides.
2.4x
< 0.3x + 6.3
Since 0.3x is added to
–0.3x
–0.3x
6.3, subtract 0.3x
from both sides.
2.1x
<
6.3
Since x is multiplied by
2.1, divide both sides
by 2.1.
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 3b Continued
Solve the inequality and graph the solutions.
Check your answer.
0.5x – 0.3 + 1.9x < 0.3x + 6
The solution set is {x:x < 3}.
x<3
–5 –4 –3 –2 –1
0
1
2
3
4
Check
Check the endpoint, 3.
2.4x – 0.3 =
2.4(3) – 0.3
7.2 – 0.3
6.9
0.3x + 6
0.3(3) + 6
0.9 + 6
6.9 
5
Check a number less
than 3.
2.4x – 0.3 < 0.3x + 6
2.4(1) – 0.3 < 0.3(1) + 6
2.4 – 0.3 < 0.3 + 6
2.1 < 6.3 
3-5
Solving Inequalities with Variables on Both Sides
Some inequalities are true no matter what value is
substituted for the variable. For these inequalities,
the solution set is all real numbers.
Some inequalities are false no matter what value is
substituted for the variable. These inequalities have
no solutions. Their solution set is the empty set, ø.
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.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 4A: All Real Numbers as
Solutions or No Solutions
Solve the inequality.
2x – 7 ≤ 5 + 2x
The same variable term (2x) appears on both
sides. Look at the other terms.
For any number 2x, subtracting 7 will always
result in a lower number than adding 5.
All values of x make the inequality true.
All real numbers are solutions.
3-5
Solving Inequalities with Variables on Both Sides
Additional Example 4B: All Real Numbers as
Solutions or No Solutions
Solve the inequality.
Distribute 2 on the left side
2(3y – 2) – 4 ≥ 3(2y + 7)
and 3 on the right side
and combine like terms.
6y – 8 ≥ 6y + 21
The same variable term (6y) appears on both sides.
Look at the other terms.
For any number 6y, subtracting 8 will never
result in a higher number than adding 21.
No values of y make the inequality true.
There are no solutions. The solution set is .
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 4a
Solve the inequality.
4(y – 1) ≥ 4y + 2
4y – 4 ≥ 4y + 2
Distribute 4 on the left side.
The same variable term (4y) appears on both sides.
Look at the other terms.
For any number 4y, subtracting 4 will never
result in a higher number than adding 2.
No values of y make the inequality true.
There are no solutions. The solution set Ø.
3-5
Solving Inequalities with Variables on Both Sides
Check It Out! Example 4b
Solve the inequality.
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.
3-5
Solving Inequalities with 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
3-5
Solving Inequalities with 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.
3-5
Solving Inequalities with Variables on Both Sides
Lesson Quiz: Part III
Solve each inequality.
5. 2y – 2 ≥ 2(y + 7)
ø
6. 2(–6r – 5) < –3(4r + 2)
all real numbers