solve compound inequalities - PMS-Math

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6.3 Students will be able to solve compound inequalities.
Warm-up

Use the numbers: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
Which numbers are less than or equal to -1
and greater than or equal to -2? -2, -1
2. Which numbers are greater than 1 or less
than -3?
-5, -4, 2, 3, 4, 5
3. Which numbers are less than or equal to -2
-5, -4, -3, -2
and less than or equal to 2?
4. Which numbers are greater than -1 or
greater than 3?
0, 1, 2, 3, 4, 5
1.
6.3 Students will be able to solve compound inequalities.
Daily Homework Quiz
For use after Lesson 6.2
Solve the inequality. Graph your solution.
1. – 72 < 8p
ANSWER
2.
p>–9
w > –5
–
–6
ANSWER
W<
– 30
6.3 Students will be able to solve compound inequalities.
EXAMPLE 1
Write and graph compound inequalities
Translate the verbal phrase into an inequality. Then
graph the inequality.
a.
All real numbers that are greater than – 2 and less
than 3.
Inequality: – 2 < x < 3
Graph:
b.
All real numbers that are less than 0 or greater than
or equal to 2.
Inequality: x < 0 or x ≥ 2
Graph:
6.3 Students will be able to solve compound inequalities.
GUIDED PRACTICE
Example 1
1.
All real numbers that are less than –1 or greater than
or equal to 4.
Inequality: x < –1 or x ≥ 4
2.
All real numbers that are greater than or equal
To –3 and less than 5.
Inequality: x ≥ –3 and x < 5 = –3 ≤ x < 5
6.3 Students will be able to solve compound inequalities.
EXAMPLE 2 Write and graph a real-world compound inequality
CAMERA CARS
A crane sits on top of a
camera car and faces toward
the front. The crane’s
maximum height and
minimum height above the
ground are shown. Write and
graph a compound inequality
that describes the possible
heights of the crane.
6.3 Students will be able to solve compound inequalities.
EXAMPLE 2 Write and graph a real-world compound inequality
SOLUTION
Let h represent the height (in
feet) of the crane. All possible
heights are greater than or
equal to 4 feet and less than
or equal to 18 feet. So, the
inequality is 4 ≤ h ≤ 18.
6.3 Students will be able to solve compound inequalities.
Solve a compound inequality with and
Solve
2 < x + 5 < 9. Graph your solution.
SOLUTION
Separate the compound inequality into two
inequalities. Then solve each inequality separately.
2 < x + 5 and
x+5<9
Write two inequalities.
2 – 5 < x + 5 – 5 and x + 5 – 5 < 9 – 5 Subtract 5 from each side.
–3 < x and
x<4
Simplify.
The compound inequality can be written as – 3 < x < 4.
6.3 Students will be able to solve compound inequalities.
EXAMPLE 3
Solve a compound inequality with and
ANSWER
The solutions are all real numbers greater than –3 and
less than 4.
Graph:
6.3 Students will be able to solve compound inequalities.
for Example 2 and 3
GUIDED PRCTICE
Investing
3.
An investor buys shares of a stock and will sell them
if the change c in value from the purchase price of a
share is less than –$3.00 or greater than $4.50. Write
and graph a compound inequality that describes the
changes in value for which the shares will be sold.
SOLUTION
Let c represent the change in the value from the
purchase price of the shares where all possible
changes are less than –$3.00 or greater than $4.50.
6.3 Students will be able to solve compound inequalities.
for Example 2 and 3
ANSWER
So the inequality is c < –3 or c > 4.5.
6.3 Students will be able to solve compound inequalities.
EXAMPLE 3
Solve a compound inequality with and
Solve the inequality. Graph your solution.
4. –7 < x – 5 < 4
SOLUTION
Separate the compound inequality into two
inequalities. Then solve each inequality separately.
–7 < x – 5 and
x–5<4
Write two inequalities.
–7 + 5 < x –5 + 5 and x – 5 + 5 < 4 + 5 Add 5 to each side.
–2 < x and
x<9
Simplify.
The compound inequality can be written as – 2 < x < 9.
6.3 Students will be able to solve compound inequalities.
EXAMPLE 3
for Example 2 and 3
ANSWER
The solutions are all real numbers greater than –2 and
less than 9.
9
Graph:
–6
–4
–2
0
2
4
6
8
10
6.3 Students will be able to solve compound inequalities.
for Example 2 and 3
GUIDED PRACTICE
Solve the inequality. Graph your solution.
5. 10 ≤ 2y + 4 ≤ 24
SOLUTION
Separate the compound inequality into two
inequalities. Then solve each inequality separately.
10 ≤ 2y + 4 and
2y + 4 ≤ 24
Write two inequalities.
10 – 4 ≤ 2y + 4 – 4 and 2y + 4 – 4 ≤ 24 – 4 Subtract 4 from each side.
6 ≤ 2y and
2y ≤ 20
Simplify.
3 ≤ y and
y ≤ 10
The compound inequality can be written as 3 ≤ y ≤ 10.
6.3 Students will be able to solve compound inequalities.
EXAMPLE 3
Solve a compound inequality with and
ANSWER
The solutions are all real numbers greater than or equal
to 3 and less than or equal to 10.
3
Graph:
0
2
4
6
8
10
12
6.3 Students will be able to solve compound inequalities.
Solve a compound inequality with and
Solve the inequality. Graph your solution.
6. –7< –z – 1 < 3
SOLUTION
Separate the compound inequality into two
inequalities. Then solve each inequality separately.
–7 < –z – 1
and –z – 1 < 3
Write two inequalities.
–7 + 1< –z – 1 + 1 and –z – 1 + 1 < 3 + 1 Add 1 to each side.
6< z
and z > – 4
Simplify.
The compound inequality can be written as – 4 < z < 6.

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