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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