Download 1-4 Solving Inequalities

1-4 Solving
Inequalities
M11.D.2.1.1: Solve compound inequalities and/or
graph their solution sets on a number line
Objectives
Solving and Graphing Inequalities
Compound Inequalities
Key Concepts
 Transitive Property
If a ≤ b and b ≤ c, then a ≤ c
Ex. if 2 ≤ 5 and 5 ≤ 11, then 2 ≤ 11
 Addition Property
If a ≤ b, then a + c ≤ b + c
If 2 ≤ 5, then 2 + 10 ≤ 5 + 10
 Subtraction Property
If a ≤ b, then a - c ≤ b – c
If 10 ≤ 15, then 10 – 2 ≤ 15 – 2
Key Concepts
 Multiplication Property
If a ≤ b and c > 0, then ac ≤ bc
If 3 ≤ 5 and 2 > 0, then 3(2) ≤ 5(2)
If a ≤ b and c < 0, then ac ≥ bc
If 3 ≤ 5 and -2 < 0, then 3(-2) ≥ 5(-2)
If you multiply by a negative number, the sign switches
Key Concepts
 Division Property
If a ≤ b and c > 0, then
If 3 ≤ 5 and 2 > 0, then
If a ≤ b and c < 0, then
If 3 ≤ 5 and -2 < 0, then
a
c
3
2
≤
≤
a
c
3

2

b
c
5
2
≥  bc
≥  25
If you multiply by a negative number, the sign switches
How to Graph Inequalities
Graph x > -3
• Step One – Mark the number with a circle
•If the inequality is > or <, don’t fill in the circle (Open)
•If the inequality is ≥ or ≤, fill in the circle (Closed)
•Step Two – Shade the line for all true values of x
• If x is greater than a number, then shade to the right.
• If x is less than a number, then shade to the left.
Another Example
Graph x ≤ 2
Solving and Graphing
Inequalities
Solve –2x < 3(x – 5). Graph the solution.
–2x < 3(x – 5)
–2x < 3x – 15
Distributive Property
–5x < –15
Subtract 3x from both sides.
x >3
Divide each side by –5 and reverse the inequality.
No Solutions or All Real Numbers
as Solutions
Solve 7x >
– 7(2 + x). Graph the solution.
7x >
– 7(2 + x)
7x >
– 14 + 7x
0>
– 14
Distributive Property
Subtract 7x from both sides.
The last inequality is always false, so 7x >
– 7(2 + x) is always false. It has
no solution.
Ex 1 & 2 as a Word Problem
A real estate agent earns a salary of $2000 per month plus
4% of the sales. What must the sales be if the salesperson is to have
a monthly income of at least $5000?
Relate: $2000 + 4% of sales >
– $5000
Define: Let x = sales (in dollars).
Write:
2000 + 0.04x >
– 5000
0.04x >
– 3000
x >
– 75,000
Subtract 2000 from each side.
Divide each side by 0.04.
The sales must be greater than or equal to $75,000.
Vocabulary
A compound inequality is a pair of inequalities
joined by and or or.
Ex. -1 < x and x ≤ 3, can also be written as -1 < x ≤ 3
Compound Inequality Containing
And
Graph the solution of 2x – 1 <
– 3x and x > 4x – 9.
2x – 1 <
– 3x and x > 4x – 9
–1 <
– x
9 > 3x
–1 <
– x and 3 > x
This compound inequality can be written as –1 <
– x < 3.
Compound Inequality Containing
Or
Graph the solution of 3x + 9 < –3 or –2x + 1 < 5.
3x + 9 < –3 or
3x < –12
–2x + 1 < 5
–2x < 4
x < –4 or x > –2
Ex. 4 & 5 as a Word Problem
A strip of wood is to be 17 cm long with a tolerance of ± 0.15
cm. How much should be trimmed from a strip 18 cm long to allow it to
meet specifications?
Relate: minimum length <
– final length <
– maximum length
Define: Let x = number of centimeters to remove.
Write:
17 – 0.15 <
– 18 – x <
– 17 + 0.15
16.85 <
– 17.15
– 18 – x <
Simplify.
–1.15 <
–
–x <
– –0.85
Subtract 18.
1.15 >
–
x >
– 0.85
Multiply by –1.
At least 0.85 cm and no more than 1.15 cm should be trimmed off to meet
specifications.
Homework
Take out a piece of new lined paper (3 hole punched)
Please put your name on the top left line and the
information below on the top right.
Pg 29 & 30
#1, 2, 14, 15, 18, 19, 22, 23, 26, 27