Linear Inequalities and Interval NotationSection 1.5Section 1.5Section 1.5

Equations and Inequalities

Interval Notation• ) or ( means “not equal to” or not inclusive

• ] or [ means “equal to” or inclusive

• ±∞ always gets a parentheses

• Written with smallest desired number on left, largest desired number on the right.– Example

EXAMPLE Graph simple inequalities

Graph x < 2.

The solutions are all real numbers less than 2.

A parenthesis is used in the graph to indicate 2 is not a solution.

)

Instead of using open/closed dots, we will now use parenthesis and brackets to indicate exclusive/inclusive. Just like interval notation.

Many times instead of using inequality symbols we will use a new notation called Interval Notation…

EXAMPLE Graph simple inequalities

Graph x ≥ –1. Interval Notation:

The solutions are all real numbers greater than or equal to –1.

A bracket is used in the graph to indicate –1 is a solution.

[

EXAMPLE Graph compound inequalities

Graph –1 < x < 2.

The solutions are all real numbers that are greater than –1 and less than 2.

( )

Interval Notation:

EXAMPLE Graph compound inequalities

Graph x ≤ –2 or x > 1.

The solutions are all real numbers that are less than or equal to –2 or greater than 1.

(]

Interval Notation: (-∞, -2] U (1, ∞)

The U means “union”…the useful values can come from either interval. Many times we take it to mean “or”

Graphing Compound Inequalities

(-∞, 5)∩(-2, ∞)

The intersection symbol ∩ means “and”. This desired result has to satisfy BOTH intervals.

Rewrite the interval as a single interval if possible.

Graph [-4,5)∩[-2,7)

Graph (-7, 3)U[0, 5)

Graph (-∞, -2]U[-2, ∞)

Rewrite in interval notation and graph

X ≤ 5

Write the inequality in interval notation

EXAMPLE Solve an inequality with a variable on one side

20 + 1.5g ≤ 50.

20 + 1.5g ≤ 50

1.5g ≤ 30

g ≤ 20

Write inequality.

Subtract 20 from each side.

Divide each side by 1.5.

ANSWER

(-∞, 20 ]

EXAMPLE Solve an inequality with a variable on both sides

Solve 5x + 2 > 7x – 4. Then graph the solution.

5x + 2 > 7x – 4

– 2x + 2 > – 4

– 2x > – 6

x < 3

ANSWERThe solutions are all real numbers less than 3. The graph is shown below.

)

(-∞, 3)

Flip the inequality when multiplying or dividing both sides by a negative #.

Solve the inequality and express in interval notation

13523 x

1555 x

31 x

42

5

3 ww

ww 54034

ww 540124

w 52

Solve and write the answer in interval notation

You rent a car for two days every weekend for a month. They charge you $50 per day, as well as $.10 per mile. Your bill has ranged everywhere from $135 to $152. What is the range of miles you have traveled?

GUIDED PRACTICE

Solve the inequality. Then graph the solution.

4x + 9 < 25

1 – 3x ≥ –14

5x – 7 ≤ 6x

3 – x > x – 9

x < 4 (-∞, 4)

ANSWER

x ≤ 5(-∞, 5]

ANSWER

x < 6(-∞, 6)

ANSWER

x > – 7[-7,∞)

ANSWER