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Simple Inequalities
Involving Addition
and Subtraction
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of contents
Remembers how to solve an algebraic equation?
x + 3 = 13
-3 -3
x = 10
Use the inverse of addition
Does 10 + 3 = 13
Be sure to13
check
= 13 your
answer!
· Solving one-step inequalities is much like solving
one-step equations.
·To solve an inequality, you need to isolate the
variable using the properties of inequalities and
inverse operations.
· Remember, whatever you do to one side, you do to
the other.
To find the solution, isolate the variable x.
Remember, it is isolated when it appears by itself on
one side of the equation.
12 > x + 5
-5
-5
Subtract to undo addition
7>x
7>x
The symbol is > so it is an open circle and it is numbers
less than 7 so it goes to the left.
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
A. j + 7 > -2
A. j + 7 > -2
-7 -7
j > -9
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-9 is not included in solution set; therefore we
graph with an open circle.
Solve and graph.
B. r - 2 > 4
r-2>4
+2 +2
r> 6
0
1
2
3
4
5
6
7
8
9
10 11
12 13 14
Solve and graph.
9>w+4
-4
-4
C. 9 > w + 4
5>w
w<5
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
128 Solve the inequality.
3<s+4
____ < s
129
Solve the inequality and graph the solution.
-4 + b < -2
A
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
C
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-4
-3
6
7
D
-10
-9
-8
-7
-6
-5
-2
-1
0
1
2
3
4
5
8
9
10
130 Solve the inequality and graph the solution.
-8 > b - 5
A
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
C
D
Simple Inequalities
Involving Multiplication
and Division
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of contents
Multiplying or Dividing by a Positive
Number
3x > -36
3x > -36
3
3
x > -12
Since x is multiplied by 3, divide
both sides by 3 for the inverse
operation.
Solve the inequality.
2 r < 4
3
( )
3
2
2 r < 4
3
r < 6
( )
3
2
Since r is multiplied by 2/3,
multiply both sides by the
reciprocal of 2/3.
132
3k > 18
A
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
C
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
D
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
133
-30 > 3q
A
10 > q
B
-10 < q
C
-10 > q
D
10 < q
134
X
< -3
2
A
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
C
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
D
136
-21 > 3d
A
d > -7
B
d > -7
C
d < -7
D
d < -7
Now let's see what happens when we multiply or
divide by negative numbers.
·Sometimes you must multiply or divide to isolate
the variable.
·Multiplying or dividing both sides of an inequality
by a negative number gives a surprising result.
Let's see what happens when we multiply this inequality
by -1.
5 > -1
-1 • 5 ? -1 • -1
We know 5 is greater than -1
Multiply both sides by -1
-5 ? 1
Is -5 less than or greater than 1?
-5 < 1
You know -5 is less than 1, so you
should use <
What happened to the inequality symbol to keep the
inequality statement true?
Helpful Hint
The direction of the inequality changes only if the
number you are using to multiply or divide by is
negative.
Solve and graph.
A. -3y > 18
Dividing each side by -3 changes
the > to <.
-3y < 18
-3
-3
y < -6
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
B.
-7m > -28
-7m < -28
-7
-7
m< 4
-10 -9
-8
-7
-6
Divide each side by -7
Flip the sign because you
divided by a negative.
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
C. 5m > -25
5m > -25
5
5
Divide each side by 5.
m > -5
-10 -9
-8
The sign does NOT change
because you did not divide by
a negative.
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-r
< 5
2
(-2)
-r
> 5
2
( )
Multiply both sides by the
reciprocal of -1/2.
-2
Why did the inequality change?
r > -10
You multiplied by a negative.
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Try these.
Solve and graph each inequality.
1. -6h < 42
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
2. 4x > -20
139
Solve and graph.
-5y ≤ -25
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
140
Solve and graph.
n>2
-2
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10