Download Chapter 4 Lesson 4

Chapter 4 Lesson 4
Additional Equations and Inequalities
Solving Radical Equations
1. Isolate a single radical on one side of the
equation
2. Square both sides of the equation (or raise
both sides to a power that is equal to the index
of the radical)
3. If a radical remains, repeat steps 1 and 2
4. Solve the resulting equation
5. All solutions must be checked in the original
equation, and only those that satisfy the
original equation are actual solutions.
Example
• Solve 𝑥 + 5 + 1 = x
Example
• Solve 4𝑥 − 8 - 1 = 2𝑥 − 5
Equation with Rational Powers
• Solve the equation(𝑥 − 3)2/3 = 4
Solving a Quadratic Inequality Algebraically
1. Write an equivalent inequality with 0 on one side
and with the function f(x) on the other side
2. Solve f(x) = 0
3. Create a sign diagram that uses the solutions from
step 2 to divide the number line into intervals. Pick a
test value in each interval and determine whether
f(x) is positive or negative in that interval to create a
sign diagram
4. Identify the intervals that satisfy the inequality in
step 1. The values of x that define these intervals are
solutions to the original inequality
Example
• Solve the inequality 𝑥 2 - 3x > 3 – 5x
Graphical Solution of Quadratic Inequality
Orientation of Graph of y
= f(x)
Inequality to Be Solved
Part of Graph That
Satisfies the Inequality
Solution to Inequality
F(x) > 0
Where the graph is above the
x-axis
x < a or x > b
F(x) < 0
Where the graph is below the
x-axis
a<x<b
F(x) > 0
Where the graph is above the
x-axis
a<x<b
F(x) < 0
Where the graph is below the
x-axis
x < a or x > b
F(x) > 0
The entire graph
All real numbers
F(x) < 0
None of the graph
No solution
F(x) > 0
None of the graph
No Solution
F(x) < 0
The entire graph
All real numbers
Height of a Model Rocket
• A model rocket is projected straight upward from ground level according to
the equation h=−16𝑡 2 +192t, t≥0 where h is height in feet and t is the time in
seconds.
• During what time interval will the height of the rocket exceed 320 feet?
Internet Use
• According to Internet World Stats, the number of worldwide Internet users,
in millions, during the years 1995 to 2007 can be modeled by the equation
y=3.604𝑥 2 +27.626x-261.714 , where x is the number of years after 1990.
• For what years from 1995 through 2015 does this model indicate that the
number of worldwide Internet users is at least 1103 million?
Power Inequalities
• To solve power inequalities, first solve the
related equation. Then use graphical methods to
find the values of the variable that satisfy the
inequality
Investment
• The future value of $3000 invested for 3 years at rate r, compounded annually, is
given by S = 3000(1 + 𝑟)3 .
• What interest rate will give a future value of at least $3630?
Inequalities Involving Absolute Value
• For a ≥ 0;
• 𝑢 < a means that –a < u < a
• 𝑢 ≤ a means that –a ≤ u ≤ a
• 𝑢 > a means that u < -a or u > a
• 𝑢 ≥ a means that u ≤ -a or u ≥ a
Example
• 2𝑥 − 3 ≤ 5
Example
• 3𝑥 + 4 - 5 > 0
Homework
• Pages 304-307
• 1,5,9,13,17,21,25,29,33,35,38,40,45,50,52,57,58
Chapter Review
• Pages 309-312
• 1-49 odd