Transcript is a solution.

Section 1.5
Linear Inequalities and Interval Notation
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 x < 2.
Graph simple inequalities
Many times instead of using inequality
symbols we will use a new notation called
Interval Notation…
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.
EXAMPLE
Graph x ≥ –1.
Graph simple inequalities
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.
Interval Notation:
The solutions are all real numbers that are greater
than –1 and less than 2.
(
)
EXAMPLE
Graph compound inequalities
Graph x ≤ –2 or x > 1.
(-∞, -2] U (1, ∞)
Interval Notation:
The U means “union”…the useful values
can come from either interval. Many times
we take it to mean “or”
The solutions are all real numbers that are less than or
equal to –2 or greater than 1.
]
(
Graphing Compound Inequalities
Rewrite the interval as a single interval if possible.
(-∞, 5)∩(-2, ∞)
The intersection symbol ∩ means “and”. This desired
result has to satisfy BOTH intervals.
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
ANSWER
(-∞, 20 ]
Write inequality.
Subtract 20 from each side.
Divide each side by 1.5.
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
Flip the inequality when
multiplying or dividing both
sides by a negative #.
(-∞, 3)
ANSWER
The solutions are all real numbers less than 3. The
graph is shown below.
)
Solve the inequality and express in
interval notation
3  2  5x  13
5  5x  15
1  x  3
Solve and write the answer in
interval notation
w3
w
2
5
4
4w  3  40  5w
4w 12  40  5w
 52  w
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
5x – 7 ≤ 6x
ANSWER
ANSWER
x<4
(-∞, 4)
x>–7
[-7,∞)
1 – 3x ≥ –14
3–x>x–9
ANSWER
ANSWER
x≤5
(-∞, 5]
x<6
(-∞, 6)