Transcript Solve Inequ w var on both sides

Solving
Inequalities
Solving
Inequalities
with with
Variables
on on
Both
Sides
Variables
Both
Sides
Warm Up
Lesson Presentation
Lesson Quiz
Holt
1 Algebra
HoltAlgebra
McDougal
McDougal
Algebra11
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
Holt McDougal Algebra 1
–4
–3
–2
–1
0
Solving Inequalities with
Variables on Both Sides
Objective
Solve inequalities that contain variable
terms on both sides.
Holt McDougal Algebra 1
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.
Holt McDougal Algebra 1
Solving Inequalities with
Variables on Both Sides
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)
Holt McDougal Algebra 1
–10 –8 –6 –4 –2
0
2
4
6
8 10
Solving Inequalities with
Variables on Both Sides
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
Holt McDougal Algebra 1
5
6
Solving Inequalities with
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
Holt McDougal Algebra 1
0
2
4
6
8 10
Solving Inequalities with
Variables on Both Sides
Check It Out! Example 1b
Solve the inequality and graph the solutions.
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
Holt McDougal Algebra 1
0
1
2
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.
3
4
5
Solving Inequalities with
Variables on Both Sides
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.
Holt McDougal Algebra 1
Solving Inequalities with
Variables on Both Sides
Example 2 Continued
Home
Cleaning
Company
siding
charge
312
plus
+
$12 per
window
12
times
# of
windows
is
less
than
Power
Clean
cost per
window
•
w
<
36
312 + 12w < 36w
– 12w –12w
312 < 24w
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.
13 < w
The Home Cleaning Company is less expensive for
houses with more than 13 windows.
Holt McDougal Algebra 1
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
•
Holt McDougal Algebra 1
Print and
# of
flyers
is less
than
More’s cost
f
<
0.25
times
# of
flyers.
per flyer
•
f
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.
Holt McDougal Algebra 1
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.
Holt McDougal Algebra 1
Solving Inequalities with
Variables on Both Sides
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
Holt McDougal Algebra 1
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
Variables on Both Sides
Example 3A Continued
–9 > k
–12
–9
Holt McDougal Algebra 1
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–3
0
3
Solving Inequalities with
Variables on Both Sides
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
To collect the variable terms,
subtract 0.4y from both sides.
0.5y ≥
– 0.5
0.5y ≥ –0.5
0.5
0.5
y ≥ –1
–5 –4 –3 –2 –1
Holt McDougal Algebra 1
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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
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
Holt McDougal Algebra 1
Solving Inequalities with
Variables on Both Sides
Check It Out! Example 3a Continued
16 ≥ 8r
Since r is multiplied by 8, divide
both sides by 8 to undo the
multiplication.
2≥r
–6
–4
Holt McDougal Algebra 1
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2
4
Solving Inequalities with
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
Simplify.
2.4x – 0.3 < 0.3x + 6
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.
x<3
Holt McDougal Algebra 1
Solving Inequalities with
Variables on Both Sides
Check It Out! Example 3b Continued
x<3
–5 –4 –3 –2 –1
Holt McDougal Algebra 1
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1
2
3
4
5
Solving Inequalities with
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.
Holt McDougal Algebra 1
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.
Holt McDougal Algebra 1
Solving Inequalities with
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)
6y – 8 ≥ 6y + 21
Distribute 2 on the left side
and 3 on the right side
and combine like terms.
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.
Holt McDougal Algebra 1
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.
Holt McDougal Algebra 1
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.
Holt McDougal Algebra 1
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
Holt McDougal Algebra 1
b < 13
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.
Holt McDougal Algebra 1
Solving Inequalities with
Variables on Both Sides
Lesson Quiz: Part III
Solve each inequality.
5. 2y – 2 ≥ 2(y + 7)
no solutions
6. 2(–6r – 5) < –3(4r + 2)
all real numbers
Holt McDougal Algebra 1