Download File

Solve the following:
(8 + v)2 – 10 = 22
Sec 6.5 Solving Inequalities Using
Addition and Subtraction
6-5 Solving Inequalities
by Adding & Subtracting
Objective
The student will be able to:
1. graph inequalities on a number line.
2. solve inequalities using addition and
subtraction.
To solve for a variable the
variable must be positive with
no number beside it!! 
x = 10
Solved
- x = -5
Not solved
2x = 5
Not solved
-3x =6
Not solved
INEQUALITIES
Another type of sentence used in algebra is called an inequality.
An inequality is used when we don't know exactly what an
expression is equal to. Instead of an equals sign, we use one of
these symbols:
This will be easier to do if you remember that the open part
of the symbol always faces the larger quantity.
MEANING
SYMBOL
<
>
≤
≥
less than
greater than
less than or equal to
greater than or equal to
Graph the solution set of
x<3
-5
0
o
5
When you have < or >, use an open dot!
Graph the solution set of
y ≥ -5/4
-5
•
0
5
When you have ≤ or ≥, use a closed dot!
Convert -5/4 to a decimal = -1.25
Graph the solution set of
p≠2
-5
0
o
5
When you have ≠, use an open dot
and shade both ways!
What is the graph of 7 ≤ m?
1.
2.
3.
4.
6
6
6
6
o
7
o
7
●
7
●
7
Answer Now
8
8
8
8
What type of dot open or closed?
1.
2.
3.
4.
5.
<
>
≤
≥
≠
Open
Open
Closed
Closed
Open
Addition & Subtraction property
of inequality
• You can add or subtract the same number or
term on each side of the inequality, the
relationship between the two sides does not
change.
y + 21 ≥ 7
x - 14 < 16
y + 21 - 21 ≥ 7 - 21
x – 14 + 14 < 16 +14
1) Solve x + (-14) < 16
x - 14 < 16
+ 14 + 14
x < 30
Solve this problem like
an equation
1. Draw “the river”
2. Eliminate double
signs
3. Add 14 to both sides
4. Simplify
5. Check your answer
6. Graph the solution
29
30 + (-14) = 16
16 = 16
o
30
31
1.
2.
2) Solve y + 21 ≥ 7
- 21 -21
Draw the “river”
Subtract 21 from
y
≥
-14
both sides
3. Simplify
4. Check your answer
5. Graph the solution
-15
30 + (-14) = 16
16 = 16
●
-14
-13
3) Solve
1.
2.
3.
4.
5.
6.
7.
8.
Draw “the river”
Subtract 8y from both
sides
Simplify
Add 14 to both sides
Simplify
Rewrite inequality
with the variable first
Check your answer
Graph the solution
16
8y + 3 > 9y - 14
- 8y
- 8y
3 > y - 14
+ 14 + 14
17 > y
y < 17
8(17) + 3 = 9(17) - 14
o
17
18
4) Solve
1.
2.
3.
4.
5.
6.
7.
Draw “the river”
Subtract 2r from both
sides
Simplify
Add 17 to both sides
Simplify
Check your answer
Graph the solution
30
3r - 17 ≥ 2r + 14
- 2r
- 2r
r – 17 ≥ 14
+ 17 + 17
r ≥ 31
3(31) - 17 = 2(31) + 14
●
31
32
Solve and graph -2x + 6 ≥ 3x - 4
-5
x≤ 2
0
5
Joanna’s tests were 87, 93, 88 and
94. What must her 5th grade be to
get a total of at least 459?
X + 87 + 93 + 88 + 94 ≥ 459
X + 362 ≥ 459
-362
-362
X ≥ 97
6-6 Solving Inequalities by
Multiplying or Dividing
Objective
The student will be able to:
solve inequalities using multiplication and
division.
POSITIVE NUMBERS ONLY
Multiplication & Division
properties of inequalities
You can multiply or divide the same positive
number or positive term on each side of the
inequality, the relationship between the two sides
does not change.
8p ≥ -96
x
< -10
3
3
x
 10 3
3
8 p
96

≥
8
8
The inequality
stays the same
NEGATIVE NUMBERS ONLY
Multiplication & Division
properties of inequalities
When multiplying or dividing the same negative
number or negative term on each side of the
inequality, you must reverse the inequality sign
x
> -10
3
3
x

10 3-3
<
3
-8p ≥ -96
8 p
96

≤
8
8
The inequality gets flipped
every time you * or / by a
negative number
To solve for a variable the
variable must be positive with
no number beside it!! 
x = 10
Solved
- x = -5
Not solved
2x = 5
Not solved
-3x =6
Not solved
1) Solve
1.
2.
3.
4.
5.
6.
Draw “the river”
Clear the fraction Multiply both sides by -4
NEW STEP!!
When multiplying BOTH
sides by a NEGATIVE
number, SWITCH the
inequality!
Simplify
Check your answer
Graph the solution
-53
4
k
>
13
4
k

13 -4
4
k < -52
52
 13
4
o
-52
-51
x
2) When solving 3 < -10
will the inequality switch?
1. Yes!
2. No!
3. I still don’t
know!
Answer Now
1.
2.
x
2) Solve
< -10
3
Draw “the river”
x
Clear the fraction 3
 10 3
Multiply both sides by 3.
3
Do you switch the
inequality?
x < -30
No - Both sides are being
multiplied by a positive
number
3.
4.
5.
30
 10
3
Simplify
Check your answer
Graph the solution
-31
o
-30
-29
a
6
3) When solving
4
will the inequality switch?
1. Yes!
2. No!
3. I still don’t
know!
Answer Now
1.
2.
a
6
3) Solve
4
Draw “the river”
a
Clear the fraction -4
 6 -4
Multiply both sides by -4.
4
Do you switch the
inequality?
a > -24
Yes - Both sides are being
multiplied by a negative
number
3.
4.
5.
(24)
6
4
Simplify
Check your answer
Graph the solution
-25
o
-24
-23
4) Solve
1.
2.
Draw “the river”
Divide both sides by -8.
Do you switch the
inequality?
Yes - Both sides are being divided
by a negative number
3.
4.
5.
Simplify
Check your answer
Graph the solution
11
-8p ≥ -96
8 p 96

8
8
p ≤ 12
-8(12) = -96
●
12
13
1) Which graph represents the
k
correct answer to
>1
4
1.
2.
3.
4.
-5
-5
-5
-5
o
-4
o
-4
●
-4
●
-4
Answer Now
-3
-3
-3
-3
Multiply both sides by -4
Switch inequality sign
K < -4
5) Solve 7v < -105
1.
2.
3.
4.
o
Divide both sides by 7
o
V < -15
-16 -15 -14
-16 -15 -14
●
-16 -15
●
-14
-15 -15 -14
Answer Now
Homework
Practice Sheet 6-5 all (9-14 show work)
Practice Sheet 6-6 all (show work for all)

Homework
Pg. 284 #8-26 Even
8-18 show work, graphing, and solution (3pts)
20-22, 26 write inequality solution, and the
graph(3 pts)
24 just graph (1pt)