graphing & solving quadratic inequalities 5.7 what is different in the graphing process of an...
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Graphing & Solving Quadratic Inequalities 5.7
What is different in the graphing process of an equality and an inequality?
How can you check the x-intercepts of a quadratic equation or inequality?
Why is it important to test 3 intervals after you have found the critical x values?
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There are four types of quadratic inequalities in two variables.
y < ax2 + bx + c
y ≤ ax2 + bx + c
y > ax2 + bx + c
y ≥ ax2 + bx + c
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Graphing a Quadratic Inequality in Two Variables
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y > x2 − 2x − 3
x = −b/2a=
y =
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Where to shade…
• Test a point inside the parabola, such as (0, 0).
• If you get a true statement when you substitute the point into the inequality, shade that area.
• If you get a false statement when you substitute the point into the inequality, shade the opposite area.
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Rappelling
A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided W ≤ 1480d 2
where d is the rope’s diameter (in inches). Graph the inequality.
SOLUTION
Graph W = 1480d 2 for nonnegative values of d. Because the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (1, 2000).
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Rappelling continued
W ≤ 1480d 2
2000 ≤ 1480(1)2
2000 ≤ 1480
Because (1, 2000) is not a solution, shade the region below the parabola.
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Solving a Quadratic Inequality by Graphing
(Looking for x intercepts)
x2 − 6x + 5 < 0.
Let y = 0 and factor to solve.
(x−1)(x−5) = 0x = 1 or x = 5
Solution 1<x<5
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Solving a Quadratic Inequality by Graphing
2x2 + 3x− 3 ≥ 0 (Use quadratic formula to factor.)
0.69 or −2.19
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Graphing a System of Quadratic Inequalitiesy ≥ x2 −4 y< −x2 −x +2
x = −b/2a =
y=
x=−b/2a =
y=
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Graph the system of quadratic inequalities.
y < – x2 + 4 Inequality 1
y > x2 – 2x – 3 Inequality 2
SOLUTION
STEP 1
Graph y ≤ – x2 + 4. The graph is the red region inside and including the parabola y = – x2 + 4.
STEP 2
Graph y > x2– 2x – 3. The graph is the blue region inside (but not including) the parabola y = x2 –2x – 3.
Identify the purple region where the two graphs overlap. This region is the graph of the system.
STEP 3
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RoboticsThe number T of teams that have participated in a robot-building competition for high school students can be modeled by
T(x) = 7.51x2 –16.4x + 35.0, 0 ≤ x ≤ 9
Where x is the number of years since 1992. For what years was the number of teams greater than 100?
T(x) > 100
7.51x2 – 16.4x – 65 > 0Graph y = 7.51x2 – 16.4x – 65 on the domain 0 ≤ x ≤ 9. The graph’s x-intercept is about 4.2. The graph lies above the x-axis when 4.2 < x ≤ 9.There were more than 100 teams participating in the years 1997–2001.
You want to find the values of x for which:
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Solve a quadratic inequality algebraicallySolve x2 – 2x > 15 algebraically.
SOLUTION
First, write and solve the equation obtained by replacing > with = .
x2 – 2x = 15
x2 – 2x – 15 = 0
(x + 3)(x – 5) = 0
x = – 3 or x = 5
Write equation that corresponds to original inequality.
Write in standard form.
Factor.
Zero product property
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The numbers – 3 and 5 are the critical x-values of the inequality x2 – 2x > 15. Plot – 3 and 5 on a number line, using open dots because the values do not satisfy the inequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.
Test x = – 4: Test x = 1:
12 –2(1) 5 –1 >15
Test x = 6:
The solution is x < – 3 or x > 5.
ANSWER
62 –2(6) = 24 >15
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What is different in the graphing process of an equality and an inequality?
•If the equation is > or <, the parabola is dashed.
•Either the inside or the outside of the parabola is shaded. Pick a point not on the parabola and see if it makes a true statement. If true, shade where the point is located. If not, shade the other area.
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What is different in the graphing process of an equality and an inequality?
How can you check the x-intercepts of a quadratic equation or inequality?
Why is it important to test 3 intervals after you have found the critical x values?
![Page 17: Graphing & Solving Quadratic Inequalities 5.7 What is different in the graphing process of an equality and an inequality? How can you check the x-intercepts](https://reader030.vdocuments.mx/reader030/viewer/2022033020/56649d935503460f94a7a545/html5/thumbnails/17.jpg)
Assignment 1.9
Page 70, 3-24 every 3rd problem, 36-57 every 3rd problem
Assignment 1.9