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© 2002 by Shawna Haider

There are two kinds of notation for graphs of inequalities: open circle or filled in circle notation and interval notation brackets. You should be familiar with both.

6 4 0 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8 [

1x

Both of these number lines show the inequality above. They are just using two different notations. Because the inequality is "greater than or equal to" the solution can equal the endpoint. That is why the circle is filled in. With interval notation brackets, a square bracket means it can equal the endpoint.

circle filled in squared end bracket

Remember---these mean the same thing---just two different notations.

6 4 0 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3

Let's look at the two different notations with a different inequality sign.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8 )

1x

Since this says "less than" we make the arrow go the other way. Since it doesn't say "or equal to" the solution cannot equal the endpoint. That is why the circle is not filled in. With interval notation brackets, a rounded bracket means it cannot equal the endpoint.

circle not filled in rounded end bracket

Remember---these mean the same thing---just two different notations.

Skill Practice

x > -1

73

c

3 > y

6 4 0 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3

Compound InequalitiesLet's consider a "double inequality"

(having two inequality signs).

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8 (

32 x

I think of these as the "inbetweeners". x is inbetween the two numbers. This is an "and" inequality which means both parts must be true. It says that x is greater than –2 and x is less than or equal to 3.

]

Skill Practice

1 42

x

0 ≤ y ≤ 8.5

Just like graphically there are three different notations, when you write your answers you can use inequality notation, set builder notation or interval notation. Again you should be familiar with both.

0 4 6 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8 [

1x {x | x≥ -1}

Inequality notation for graphs shown above.

),1[

Interval notation for graphs shown above.

Set builder notation for graphs shown above.

),1[

Let's have a look at the interval notation.

For interval notation you list the smallest x can be, a comma, and then the largest x can be so solutions are anything that falls between the smallest and largest.

This means x is unbounded

above

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8 [

unbounded

The bracket before the –1 is square because this is greater than "or equal to" (solution can equal the endpoint).

The bracket after the infinity sign is rounded because the interval goes on forever (unbounded) and since infinity is not a number, it doesn't equal the endpoint (there is no endpoint).

]4,2(Let's try another one.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Rounded bracket means cannot equal -2

Squared bracket means can equal 4

The brackets used in the interval notation above are the same ones used when you graph this.

( ]

This means everything between –2 and 4 but not including -2

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

)4,(Let's look at another one

This means x is unbounded

below

Notice how the bracket notation for graphing corresponds to the brackets in interval notation.

Remember that square is "or equal to" and round is up to but not equal. By the infinity sign it will always be

round because it can't equal infinity (that is not a number).

This means the largest x can be is 4

but can't equal 4

)

Addition and Subtraction Properties of Inequality.Let a, b, and c represent real numbers1. Addition Property of Inequality: If a < b,

Then a + c < b + c

2. Subtraction Property of inequality If a < bThen a – c < b - c

SOLVING A LINEAR INEQUALITY

-2p + 5 < -3p + 6

-2p + 5 < -3p + 6-2p + 5 +3p < -3p +3p + 6

Addition property of inequality (add 3p to both sides).

Simplifyp + 5 < 6

Subtraction property of inequality.p + 5 - 5 < 6 – 5p < 1

Set-Builder notation: {p | p < 1}Interval notation: (-∞, 1)

Properties of Inequalities.

Essentially, all of the properties that you learned to solve linear equations apply to solving linear inequalities with the exception that if you multiply or divide by a negative you must reverse the inequality sign.

So to solve an inequality just do the same steps as with an equality to get the variable alone but if in the process you multiply or divide by a negative let it ring an alarm in your brain that says "Oh yeah, I have to turn the sign the other way to keep it true".

Example: 8462 xx- 4x - 4x

862 x + 6 +6

142 x -2 -2

Ring the alarm! We divided by a

negative!

7xWe turned the sign!

Skill Practice Solving a Linear Inequality

-5x -3 ≤ 12 Add 3 to both sides

-5x -3 +3 ≤ 12 +3-5x ≤ 15

Divide by -5. Reverse the direction of the inequality sign.

-5x ≥ 15-5 -5

x ≥ -3 or [-3, ∞)

Solving a Compound Inequality of the Form

a < x < b-3 ≤ 2x + 1 < 7 Subtract -1 from all three

parts of the inequality-3 -1 ≤ 2x +1 -1 < 7 -1 -4 ≤ 2x < 6 Divide by 2 in all three

parts of the inequality-4 ≤ 2x < 6 2 2 2 -2 ≤ x < 3 or [ -2, 3)

Applications of Linear Inequalities

English Phrase Mathematical Inequality

a is less than b a < b

a is greater than ba exceeds b

a > b

a is less than or equal to b a is at most ba is no more than b

a ≤ b

a is greater than or equal to ba is at least ba is no less than b

a ≥ b

Translating Expressions Involving Inequalities

Bill needs a score of at least 92 on the final exam. Let x represent Bill’s score x ≥ 92

Fewer than 19 cars are in the parking lot. Let c represent the number of cars.c < 19

The heights, h, of women who wear petite sixe clothing are typically between 58 in. and 63 in, inclusive.

58 ≤ h ≤ 63

Solving an Application with Linear Inequalities

To earn an A in a math class, Alsha must average at least 90 on all of her tests. Suppose Alsha has scored 79, 86, 93, 90, and 95 on her first five math tests. Determine the minimum score she needs on her sixth test to get an A in the class

x ≥ 97