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Solve Inequality Notes
Question
How do you graph
inequalities?
Answer
1. Solve for the variable.
x+3<5
x < 2 Subtract 3 from both sides
2. Look at inequality sign to determine if you use a open or closed dot.
 or : Use closed dot
because it includes the value
> or <: Use open dot
because it does not include value
3. Place appropriate dot over value on number line.
-3
-2
-1
0
1
2
3
4
4. Test a value on either side of the number line to determine which way to draw
the arrow for the solution set.
* Test value to right of dot: Pick a number, say 4. Is 4 less than 2? No
* Test value to left of dot: Pick a number, say 0. Is 0 less than 2?
Yes, so arrow points to the left.
5. Draw arrow on number line to indicate solution set for equation.
How do you write
an inequality from
a graph?
-3
-2
-1
0
1
2
3
4
1. Pick a variable for your inequality, say x.
2. Determine the inequality sign by looking at the dot.
In the example, it is a closed dot so x will be  or .
3. Pick one of the inequality signs. Place the variable on one side and the value
under the dot on the other. For example, x  -1
4. Test a value to check if the right inequality sign is chosen.
For example: Pick 2. Is 2  -1? Yes, so you chose the right sign.
How do you solve
inequalities using
addition and
subtraction?
How do you solve
inequalities using
multiplication and
division?
Add or Subtract the opposite to solve for variable.
d + 5 > 14
d + 5 – 5 >14 – 5 (Subtract 5 since it is the opposite of adding 5)
d>9
(Combine like terms)
Positive Number: When multiplying or dividing by a positive number, the
inequality sign stays the same.
2x < 8
2x < 8
2
2
x<4
Problem
Divide each side by 2 to solve for x
Simplify
Solve Inequality Notes
Negative Number: When multiplying or dividing by a negative, change the
inequality sign.
-x4
2
-2 ∙ - x  4 ∙ (-2) Opposite of division is multiplication, so multiply each
2
Side by 2.
x  -8
How do you solve
multi-step
inequalities?
Simplify and change sign because you multiplied by a
negative number.
Example Problem: 4(3m – 1) > 6m + 2m + 15
1. Simplify each side before moving any terms. Simplify by:
- Combining like terms(The like terms must be on the same side of the
inequality side before you can combine). Use properties of Inequalities to
move terms)
- Using the Distributive Property
4(3m – 1) > 6m + 2m + 15
12m – 4 > 8m + 15
(Simplified each side separately)
2. Pick a side for the variable.
Do addition and subtraction first to move terms.
Solve Compound
Inequalities
12m – 4 > 8m + 15
12m – 4 + 4 > 8m + 15 + 4
(Add 4 to each side)
12m > 8m + 19
( Simplify by combining like terms)
12m - 8m > 8m + 19 – 8m
( Subtract 8m from both sides)
4m > 19
(Simplify by combining like terms)
4m > 19
( Divide each side by 4 – Opposite of
4
4
multiplication)
m>4¾
(Simplify)
Same as Solving Multistep only with sections
6 < x + 2 < 12
-2
-2
-2
0
4
10
4 < x < 10
2x – 3 < 7 or 3 – x < -8 Solve each inequality for the variable.
X<5 or x > 11
Reminder: If you multiple/divide by negative
change the inequality sign
Absolute Values
0 5
11 14
|2x + 1| > 7
2x + 1 > 7 or 2x + 1 >7 Rewrite as two statements
2x + 1 >7 or 2x + 1 < -7 On the second one flip the inequality and negate 7
x > 3 or x < - 4
- 4
0
3
Solve Inequality Notes