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Inequalities
LT 1.5
Remember?!
1) Think of a instance where you dealt with inequalities. It may be a
instance, a circumstance or maybe something that you remember
from the unit of inequalities.
2) Talk to your elbow partner and recall a memory from your past that
has to do with inequalities.
3) Be ready to share!
We have used inequalities in
geometry!
Pythagorean inequality theorem states that:
If 𝑎 2 + 𝑏 2 > 𝑐 2 then the triangle is Acute
If 𝑎 2 + 𝑏 2 < 𝑐 2 then the triangle is obtuse.
If 𝑎 2 + 𝑏 2 = 𝑐 2 then the triangle is right.
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Acute, obtuse or right? You
try!
Inequality Review
Inequalities act just like equations, but there are a few exceptions
1) ÷ 𝑜𝑟 ∙ by a negative does what?
2) Compound inequalities are used
3) you have multiple answers that satisfy an inequality.
Problems
Solve the following inequalities.
a) 2𝑥 + 1 ≤ 5
B) 2 − 6𝑥 > 5𝑥 + 7
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Inequalities Involving Absolute
Value
The rules:
1. |𝑥| < 𝑎 if and only if 𝑥 > −𝑎 and 𝑥 < 𝑎 (i.e., −𝑎 < 𝑥 < 𝑎)
2. |𝑥| > 𝑎 if and only if 𝑥 < −𝑎 or 𝑥 > 𝑎
Graphically, this is what happens:
1) 𝑥 < 𝑎
2) 𝑥 > 𝑎
Interval Notation: New stuff
• Interval notation is a way of identifying a solution set.
•The solution set of an inequality is the set of all solutions of the inequality.
For a inequality: 𝑥 ≥ 4
For interval notation [4, ∞)
•Lets break it down!
>, < symbols are ( ) symbols, They do not include the number. Graphically, there
are holes as endpoints.
≥, ≤ symbols are symbols, They include the number. Graphically, there are filled
in holes as endpoints.
Examples
Represent in interval notation and graph:
1) 𝑥 > 3
2) 𝑥 ≤ −2
3) −3 ≤ 𝑥 < 10
Mind Blown…
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More Complex Problems
Solve each inequality. Show your answer in both interval notation
and inequality notation.
a) 2 − 𝑥 < 5
b) 𝑥 + 4 ≥ 2
Review Questions
LT 1.1-1.5
Review Questions
Problem 1. Find the standard form of the equation of the
circle with center at (2, -5) and radius 4.
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Review Questions
Problem 2: Use a graphing utility and proper window settings to graph
𝑥 3 + 𝑦 – 2𝑥 = 0.
Review Questions
Problem 3. A company purchases a $20,000 machine. In 4 years the
machine will be worth $10,000. Write a linear equation that relates
the value V of the machine after t years.
Review Questions
Problem 4. Solve:
𝑥
𝑥−1
=
1
1
−
𝑥−1
𝑥−3
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Review Questions
Problem 5: Find the x- and y-intercepts of the graph of each equation.
a) 2𝑥 + 3𝑦 = 6
b) 𝑦 = 𝑥 2 + 𝑥 – 6
Review Questions
Problem 6: Approximate the points of intersection of the graphs of the
following equations.
𝑦 = 𝑥 2 + 2𝑥 – 8
𝑦 = 𝑥 3 + 𝑥 2 – 6𝑥 + 2
Review Questions
Problem 7: Solve by extracting square roots.
16𝑥 2 = 25
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Review Questions
Problem 8. Solve by completing the square.
𝑥 2 + 4𝑥 = 5
Review Questions
Problem 9. Solve by using the Quadratic Formula.
3𝑥 2 − 𝑥 − 5 = 0
Review Questions
Problem 10: Solve
𝑥2 − 6 = 𝑥
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Review Questions
Problem 11. Solve
a) 𝑥 3 = 9𝑥
b) 𝑥 3 − 𝑥 2 − 4𝑥 + 4 = 0
More to do!
Problem 12. Solve
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a) 𝑥 2 − 27 = 0
b) 𝑥 − 3 + 5 = 0
A Hard one to Remember
Problem 13: Solve the equation for x.
𝑥−5 + 𝑥+7= 6
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You Try Problem
𝑥 + 𝑥 − 20 = 10
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