Transcript 5.8 – Solving Quadratic Inequalities

5.8 – Solving Quadratic
Inequalities
Objectives:
Write, solve, and graph a quadratic inequality in one variable.
Write, solve, and graph a quadratic inequality in two variables.
Standard:
2.8.11.H. Select and use an appropriate strategy to solve
inequalities.
I. One-Variable Quadratic Inequalities
You can determine the solution to a given inequality by finding the roots of the
related quadratic equation or by using the graph of the related quadratic
equation.
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Less ThAND GreatOR
1b.
x2 – 8x + 12 ≤ 0
x ≥ smaller root and x ≤ larger root
x2 – 8x + 12 = 0
(x – 6)(x – 2) = 0
x = 6 and x = 2
Therefore, x ≥ 2 and x ≤ 6
Katie makes and sells T-shirts. A consultant found that her monthly
costs, C, are related to the selling price, p, of the shirts by the
function C(p) = 75p + 2500. The revenue, R, from the sale of the
shirts is represented by R(p) = -25p2+ 700p. Her profit, P, is the
difference between the revenue and the costs each month. P(p) =
R(p) – C(p)
= -25p2+ 700p – (75p + 2500)
= -25p2+ 625p – 2500
At what price range can Katie sell her T-shirts in order to make
a profit?
-25p2 + 625p – 2500 > 0
Divide both sides by -25
p2 – 25p + 100 < 0
Factor
(p – 20)(p – 5) < 0
p < 20 and p > 5
Ex 3. Solve each inequality
II. Two-Variable Quadratic
Inequalities
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A quadratic inequality in two variables is
an inequality that can be written in one of the
forms below, where a, b, and c are real
numbers and a ≠ 0.
y ≥ ax2+ bx + c
y > ax2+ bx + c
y ≤ ax2+ bx + c
y < ax2+ bx + c
b. y < (x – 1)2 – 5
c. y ≤ (x + 2)2 - 3