INEQUALITIESINEQUALITIES

Math 7Math 7

SymbolsSymbols

< less than< less than

> greater than> greater than

≤ ≤ less than or equal toless than or equal to

≥ ≥ greater than or equal togreater than or equal to

Meaning of SymbolsMeaning of Symbols

x > 2 means that the value of x is greater x > 2 means that the value of x is greater than 2 (but does not include 2)than 2 (but does not include 2)

x ≥ 2 means that the value of x is greater x ≥ 2 means that the value of x is greater than or equal to 2 (includes 2 and than or equal to 2 (includes 2 and anything larger)anything larger)

x < 2 means that the value of x is less x < 2 means that the value of x is less than 2 (but does not include 2)than 2 (but does not include 2)

x ≤ 2 means that the value of x is less x ≤ 2 means that the value of x is less than or equal to 2 (includes 2 and than or equal to 2 (includes 2 and anything smaller)anything smaller)

ExamplesExamples

x > 4 What could x be?x > 4 What could x be? x < 9 What could x be?x < 9 What could x be? x ≤ 7 What could x be?x ≤ 7 What could x be? x ≥ 6 What could x be?x ≥ 6 What could x be? x < -8 What could x be?x < -8 What could x be? x > -13 What could x be?x > -13 What could x be? x ≥ -9 What could x be?x ≥ -9 What could x be? x ≤ -4 What could x be?x ≤ -4 What could x be?

INEQUALITIES IN INEQUALITIES IN WORDSWORDS

Sometimes, you will have to read a Sometimes, you will have to read a statement to determine which inequality to statement to determine which inequality to use.use.

Read the statement carefully because these Read the statement carefully because these can get tricky!can get tricky!

When you see words like “at least” or “at When you see words like “at least” or “at most”, how do you handle them?most”, how do you handle them? For example: At least 5 people were late todayFor example: At least 5 people were late today

The stadium holds at most 48,000 peopleThe stadium holds at most 48,000 people

BE CAREFUL!BE CAREFUL!

No more than 50 studentsNo more than 50 students

Fewer than 12 itemsFewer than 12 items

Up to and including 12Up to and including 12

*The speed limit on a road is 55 *The speed limit on a road is 55 mph*mph*

BACKWARDS?BACKWARDS?

x > 9 is read “x is greater than 9”x > 9 is read “x is greater than 9” What does this say about 9?What does this say about 9?

So, if you had 12 > x, what does that So, if you had 12 > x, what does that say about x?say about x?

If they are backwards, you don’t If they are backwards, you don’t have to panic!have to panic!

13 < x would mean what?13 < x would mean what?

TRANSLATE SOMETRANSLATE SOME

the number of students is greater than 13the number of students is greater than 13 the amount of money is less than or equal the amount of money is less than or equal

to $50to $50 60 is greater than or equal to the number60 is greater than or equal to the number the perimeter is no less than 85 inchesthe perimeter is no less than 85 inches the area is no greater than 100 square cmthe area is no greater than 100 square cm fewer than 12 itemsfewer than 12 items at least 275 people attended the playat least 275 people attended the play at most $50 for admission to the gameat most $50 for admission to the game

Graphing InequalitiesGraphing Inequalities

These are graphed on number lines. These are graphed on number lines. Be sure the problem is solved for x.Be sure the problem is solved for x. Draw a number line with x, some values Draw a number line with x, some values

smaller than x, and some larger.smaller than x, and some larger. Go to x on the number line and use:Go to x on the number line and use:

○ ○ if the number “is not included in the if the number “is not included in the solution” (symbol is < or >)solution” (symbol is < or >)

● ● if the number “is included in the if the number “is included in the solution” (symbol is ≤ or ≥)solution” (symbol is ≤ or ≥)

Graphing InequalitiesGraphing Inequalities

The last step is to draw an arrow The last step is to draw an arrow pointing in the proper direction.pointing in the proper direction.““less than” should be pointed in direction less than” should be pointed in direction

that has numbers less than xthat has numbers less than x

““greater than” should be pointed in the greater than” should be pointed in the direction that has numbers greater than direction that has numbers greater than xx

ExamplesExamples

x > 5x > 5

x < 9x < 9

x ≥ 6x ≥ 6

x ≤ 3x ≤ 3

ExamplesExamples

x < -7x < -7

x > -3x > -3

x ≤ -5x ≤ -5

x ≥ -8x ≥ -8

CAREFUL WITH THE CAREFUL WITH THE BACKWARDS EXAMPLESBACKWARDS EXAMPLES 9 < x9 < x

-11 > x-11 > x

7 ≥ x7 ≥ x

-3 ≤ x-3 ≤ x

Solving One-Step Solving One-Step InequalitiesInequalities

Sometimes these will not be in the form “x Sometimes these will not be in the form “x > ..”> ..”

In this case, we will have to solve for x. In this case, we will have to solve for x. Example: x – 9 > 5Example: x – 9 > 5

Solve for x as if the > were an = (inverse Solve for x as if the > were an = (inverse operations). operations).

Check to make sure it creates a true Check to make sure it creates a true statementstatement

When you check, make sure to choose a When you check, make sure to choose a number that is described by the inequalitynumber that is described by the inequality

ExamplesExamplesCHECKSCHECKS

x – 9 < 11x – 9 < 11

x + 12 > 9x + 12 > 9

12 ≥ -18 + x12 ≥ -18 + x

-11 ≥ x – 18-11 ≥ x – 18

TRANSLATE, SOLVE , TRANSLATE, SOLVE , GRAPHGRAPH

The sum of t and 9 is greater than or equal The sum of t and 9 is greater than or equal to 36to 36

7 more than w is less than or equal to 107 more than w is less than or equal to 10

19 increased by p is greater than or equal to 19 increased by p is greater than or equal to 22

Three plus a number is at least sixThree plus a number is at least six

ExamplesExamples

5x > -405x > -40

3x < 123x < 12

5x ≤ -305x ≤ -30

7x ≤ -567x ≤ -56

72

x

78

x

103

2x

225

12 x

DIVIDING BY A DIVIDING BY A NEGATIVENEGATIVE

When solving, dividing by a negative When solving, dividing by a negative changes the game of the inequality. changes the game of the inequality.

Let’s see what happens here:Let’s see what happens here: -7x > 63-7x > 63 To solve, you must divide by -7 on To solve, you must divide by -7 on

each sideeach side x > -9x > -9 Choose a number to plug this Choose a number to plug this

back back into the original to see if it into the original to see if it “checks”“checks”

-7(-8) > 63? Is this true? Ok, try -7 (since -7 -7(-8) > 63? Is this true? Ok, try -7 (since -7 > -9)> -9)

-7(-7) > 63?-7(-7) > 63? Is this true? Is this true? So try something smaller than -9…like -10. So try something smaller than -9…like -10.

DIVIDING BY A DIVIDING BY A NEGATIVENEGATIVE

-10 did work…weird right?-10 did work…weird right? Basically, whenever you divide by a negative, Basically, whenever you divide by a negative,

the inequality needs to be flipped around to the inequality needs to be flipped around to the opposite (less than goes to greater than the opposite (less than goes to greater than and vice versa; less than or equal to goes to and vice versa; less than or equal to goes to greater than or equal to and vice versa)greater than or equal to and vice versa)

This only happens when the number you This only happens when the number you are DIVIDING BY is negativeare DIVIDING BY is negative

This will also happen if you multiply This will also happen if you multiply both sides by a negative (like when we both sides by a negative (like when we use the multiplicative inverse to clear a use the multiplicative inverse to clear a fraction)fraction)

MORE EXAMPLESMORE EXAMPLES

Solve and Graph:Solve and Graph: -9x > 90-9x > 90

-11x < -99-11x < -99

12x ≤ -3612x ≤ -36

-5x ≥ 100-5x ≥ 100

72

x

TRANSLATE, SOLVE , TRANSLATE, SOLVE , GRAPHGRAPH

Double a number is at most fourDouble a number is at most four

6 is less than the product of f and 6 is less than the product of f and 2020

A number divided by -3 is at least A number divided by -3 is at least sevenseven

TWO-STEP TWO-STEP INEQUALITIESINEQUALITIES

Treat these the same way you would Treat these the same way you would treat a two-step equationtreat a two-step equation

Take care of the addition/subtraction Take care of the addition/subtraction part firstpart first

Then move on to the Then move on to the multiplication/division portionmultiplication/division portion

Remember, if you multiply or divide Remember, if you multiply or divide BY a negative, you must flip the BY a negative, you must flip the symbolsymbol

SOLVE THE SOLVE THE INEQUALITIESINEQUALITIES

5x – 4 > -245x – 4 > -24

6 + 3x < -276 + 3x < -27

-9y + 33 ≥ 51-9y + 33 ≥ 51