INEQUALITIES

Targeted TEKS:

A.10 The student understands there is more than one way to solve a Quadratic Equation and solves them using appropriate methods.

(A) Solve Quadratic Equations using concrete models, tables, graphs, and algebraic methods

Equal or Unequal?

• We call a math statement an EQUATION when both sides of the statement are equal to each other.– Example: 10 = 5 + 3 + 2

• We call a math statement an INEQUALITY when both sides of the statement are not equal to each other.– Example: 10 = 5 + 5 + 5

Inequality Signs

• We don’t use the = sign if both sides of the statement are not equal, we use other signs.

GREATER THAN GREATER THAN (OR EQUAL TO)

LESS THAN LESS THAN (OR EQUAL TO)

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DON’T FORGET THIS!!!

• THE BIGGER SIDE OF THE SIGN IS ON THE SAME SIDE AS THE BIGGER #

• THE SMALLER SIDE OF THE SIGN IS ON THE SAME SIDE AS THE SMALLER #

– Examples: 10 15 or -4 -12< >

Let’s Try Some!

• 3 5

• 22 10

• -10 4

• 2 7

• -65 -62

• 32.332.5

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Our Friend, The Number Line

• A number line is simply this…

…a line with numbers on it.

• We use a number line to count and to graphically show numbers.– Example: Graph x = 5.

Graphing Inequalities

• Graph x = 2

• Graph x < 2

• Graph x < 2

• Graph x > 2

• Graph x > 2

A “closed” circle ( )indicates we includethe number.

An “open” circle ( )indicates we DO NOTinclude the number.

By shading in the number line we areindicating that all thenumbers in the shadeare also possible answers.

You Try This…

• Graph x < 10

You Try This…

• Graph x > -4

You Try This…

• Graph x > 200

You Try This…

• Graph 7 < x

Let’s Go Shopping!

• Last week you went shopping at the mall. You had $150 to spend for the day. You bought a shirt for $25 and some jeans for $40. You also spent $5 on lunch. You wanted to purchase a pair of shoes. What is the maximum amount of money you could have spent on the shoes?

$150 > $25 + $40 + $5 + xThe maximum amount you have

The amount youhave spent

The cost ofthe shoes

How much can the shoes cost?

• Basically, the shoes must cost less than or equal to the amount you have left!

$150 > $25 + $40 + $5 + x

$150 > $70 + x-$ 70 -$70$ 80 > x

The cost ofthe shoes

Do You Really Understand?

• Let’s see if this makes sense…

(If we add 6 to both sides, is the inequality true?)

3 < 9

3+6 < 9+6

9 < 15

YES!

Do You Really Understand?

• Let’s see if this really makes sense…

(If we subtract 3 from both sides, is the inequality true?)

10 > 4

10-3 > 4-3

7 > 1

YES!

Do You Really Understand?

• Let’s see if this still really makes sense…

(If we multiply both sides by 2, is the inequality true?)

8 < 12

8(2) < 12(2)

16 < 24

YES!

Do You Really Understand?

• Let’s see if this still really makes sense…

(If we multiply both sides by -2, is the inequality true?)

8 < 12

8(-2) < 12(-2)

-16 < -24THIS STATEMENTIS NOT TRUE. WENEED TO FLIP THEINEQUALITY SIGNTO MAKE THIS ATRUE STATEMENT.

-16 > -24

Solving Inequalities

• So apparently there are a few basic rules we have to follow when solving inequalities.

• If you break these rules you will answer the question incorrectly!

• DON’T BREAK THE RULZ!

Rule #1

• Don’t forget who the bigger number is!– Example:

9 > x

– It is okay to rewrite this statement as

x < 9

– If 9 is bigger than “x”, that means that “x” is smaller than 9.

Rule #2

• When multiplying or dividing by a negative number, reverse the inequality sign.– Example:

15 > -5x-5 -5

-3 < x

Solve Each Inequality & Graph

Example 1:

m + 14 < 4-14 -14

m < -10

Solve Each Inequality & Graph

Example 2:

6y - 6 > 7y-6y -6y -6 > y

y < -6

Solve Each Inequality & Graph

Example 3:

k < 10-3

(-3) (-3)

k > -30