# INEQUALITIES

```INEQUALITIES
Math 7
Symbols

&lt; less than

&gt; greater than

≤ less than or equal to

≥ greater than or equal to
Meaning of Symbols




x &gt; 2 means that the value of x is greater than 2
(but does not include 2)
x ≥ 2 means that the value of x is greater than or
equal to 2 (includes 2 and anything larger)
x &lt; 2 means that the value of x is less than 2
(but does not include 2)
x ≤ 2 means that the value of x is less than or
equal to 2 (includes 2 and anything smaller)
Examples








x &gt; 4 What could x be?
x &lt; 9 What could x be?
x ≤ 7 What could x be?
x ≥ 6 What could x be?
x &lt; -8 What could x be?
x &gt; -13 What could x be?
x ≥ -9 What could x be?
x ≤ -4 What could x be?
INEQUALITIES IN WORDS



Sometimes, you will have to read a statement to
determine which inequality to use.
Read the statement carefully because these can
get tricky!
When you see words like “at least” or “at most”,
how do you handle them?

For example: At least 5 people were late today

The stadium holds at most 48,000 people
BE CAREFUL!

No more than 50 students

Fewer than 12 items

Up to and including 12

*The speed limit on a road is 55 mph*
BACKWARDS?





x &gt; 9 is read “x is greater than 9”
What does this say about 9?
So, if you had 12 &gt; x, what does that say about
x?
If they are backwards, you don’t have to panic!
13 &lt; x would mean what?
TRANSLATE SOME








the number of students is greater than 13
the amount of money is less than or equal to \$50
60 is greater than or equal to the number
the perimeter is no less than 85 inches
the area is no greater than 100 square cm
fewer than 12 items
at least 275 people attended the play
at most \$50 for admission to the game
Graphing Inequalities
These are graphed on number lines.
 Be sure the problem is solved for x.
 Draw a number line with x, some values smaller
than x, and some larger.
 Go to x on the number line and use:
○ if the number “is not included in the solution”
(symbol is &lt; or &gt;)
● if the number “is included in the solution”
(symbol is ≤ or ≥)

Graphing Inequalities

The last step is to draw an arrow pointing in the
proper direction.
“less than” should be pointed in direction that has
numbers less than x
“greater than” should be pointed in the direction that
has numbers greater than x
Examples

x&gt;5

x&lt;9

x≥6

x≤3
Examples

x &lt; -7

x &gt; -3

x ≤ -5

x ≥ -8
CAREFUL WITH THE
BACKWARDS EXAMPLES

9&lt;x

-11 &gt; x

7≥x

-3 ≤ x
Solving One-Step Inequalities






Sometimes these will not be in the form “x &gt; ..”
In this case, we will have to solve for x.
Example: x – 9 &gt; 5
Solve for x as if the &gt; were an = (inverse
operations).
Check to make sure it creates a true statement
When you check, make sure to choose a number
that is described by the inequality
Examples
CHECKS

x – 9 &lt; 11

x + 12 &gt; 9

12 ≥ -18 + x

-11 ≥ x – 18
TRANSLATE, SOLVE , GRAPH

The sum of t and 9 is greater than or equal to 36

7 more than w is less than or equal to 10

19 increased by p is greater than or equal to 2

Three plus a number is at least six
Examples

5x &gt; -40

3x &lt; 12

5x ≤ -30

7x ≤ -56
x
7
2
x
 7
8
2
x  10
3
1
2 x  22
5
DIVIDING BY A NEGATIVE


When solving, dividing by a negative changes
the game of the inequality.
Let’s see what happens here:
To solve, you must divide by -7 on
each side
 x &gt; -9
Choose a number to plug this back
into the original to see if it “checks”
 -7(-8) &gt; 63? Is this true? Ok, try -7 (since -7 &gt; -9)
 -7(-7) &gt; 63? Is this true?
 So try something smaller than -9…like -10.

-7x &gt; 63
DIVIDING BY A NEGATIVE




-10 did work…weird right?
Basically, whenever you divide by a negative, the
inequality needs to be flipped around to the opposite
(less than goes to greater than and vice versa; less than
or equal to goes to greater than or equal to and vice
versa)
This only happens when the number you are
DIVIDING BY is negative
This will also happen if you multiply both sides
by a negative (like when we use the
multiplicative inverse to clear a fraction)
MORE EXAMPLES

Solve and Graph:
-9x &gt; 90

-11x &lt; -99

12x ≤ -36

-5x ≥ 100

x
7
2
TRANSLATE, SOLVE , GRAPH

Double a number is at most four

6 is less than the product of f and 20

A number divided by -3 is at least seven
TWO-STEP INEQUALITIES




Treat these the same way you would treat a twostep equation
Take care of the addition/subtraction part first
Then move on to the multiplication/division
portion
Remember, if you multiply or divide BY a
negative, you must flip the symbol
SOLVE THE INEQUALITIES

5x – 4 &gt; -24

6 + 3x &lt; -27

-9y + 33 ≥ 51
```