graphing quadratic inequalities

Graphing Quadratic Inequalities

4.9

EXAMPLE 1 Graph a quadratic inequality

Graph y > x2 + 3x – 4.

SOLUTION

STEP 1

Graph y = x2 + 3x – 4. Because the inequality symbol is >, make the parabola dashed.

Test a point inside the parabola, such as (0, 0).

STEP 2

y > x2 + 3x – 4

0 > 02 + 3(0) – 4?

0 > – 4

EXAMPLE 1 Graph a quadratic inequality

So, (0, 0) is a solution of the inequality.

STEP 3

Shade the region inside the parabola.

EXAMPLE 2 Use a quadratic inequality in real life

A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided

Rappelling

W ≤ 1480d2

where d is the rope’s diameter (in inches). Graph the inequality.

SOLUTION

Graph W = 1480d2 for nonnegative values of d. Because the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (1, 2000).

EXAMPLE 2 Use a quadratic inequality in real life

W ≤ 1480d2

2000 ≤ 1480(1)2

2000 ≤ 1480

Because (1, 2000) is not a solution, shade the region below the parabola.

EXAMPLE 3 Graph a system of quadratic inequalities

Graph the system of quadratic inequalities.

y < – x2 + 4 Inequality 1y > 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.

EXAMPLE 3 Graph a system of quadratic inequalities

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

GUIDED PRACTICE for Examples 1, 2, and 3

Graph the inequality.

1. y > x2 + 2x – 8

STEP 1

Graph y = x2 + 2x – 8. Because the inequality symbol is >, make the parabola dashed.


STEP 2

y > x2 + 2x – 8

0 > 02 + 2(0) – 8?

0 > – 4



STEP 3




y < 2x2 – 3x + 12.

SOLUTION

STEP 1

Graph y = 2x2 – 3x + 1. Because the inequality symbol is <, make the parabola dashed.


STEP 2

y < 2x2 – 3x + 1

0 < 02 – 3(0) + 1?

0 < 1



STEP 3




y < – x2 + 4x + 23. SOLUTION

STEP 1

Graph y = – x2 + 4x + 2. Because the inequality symbol is <, make the parabola dashed.


STEP 2

y < – x2 + 4x + 2

0 < 02 + 4(0) + 2?

0 < 2



STEP 3



4. Graph the system of inequalities consisting of y ≥ x2 and y < 2x2 + 5.

SOLUTION

STEP 1

STEP 2

Graph y < 2x2 + 5.

Graph y > x2.–

Identify the shaded region where the two graphs overlap. This region is the graph of the system.

STEP 3

EXAMPLE 4 Solve a quadratic inequality using a table

Solve x2 + x ≤ 6 using a table.

SOLUTION

Rewrite the inequality as x2 + x – 6 ≤ 0. Then make a table of values.

Notice that x2 + x –6 ≤ 0 when the values of x are between –3 and 2, inclusive.

The solution of the inequality is –3 ≤ x ≤ 2.

ANSWER

EXAMPLE 5 Solve a quadratic inequality by graphing

Solve 2x2 + x – 4 ≥ 0 by graphing.

SOLUTION

The solution consists of the x-values for which the graph of y = 2x2 + x – 4 lies on or above the x-axis. Find the graph’s x-intercepts by letting y = 0 and using the quadratic formula to solve for x.

0 = 2x2 + x – 4

x = – 1+ 12– 4(2)(– 4)2(2)

x =– 1+ 33

4

x 1.19 or x –1.69

EXAMPLE 5 Solve a quadratic inequality by graphing

Sketch a parabola that opens up and has 1.19 and –1.69 as x-intercepts. The graph lies on or above the x-axis to the left of (and including) x = – 1.69 and to the right of (and including) x = 1.19.

The solution of the inequality is approximately x ≤ – 1.69 or x ≥ 1.19.

ANSWER

GUIDED PRACTICE for Examples 4 and 5

Solve the inequality 2x2 + 2x ≤ 3 using a table and using a graph. 5.

SOLUTION

Rewrite the inequality as 2x2 + 2x – 3 ≤ 0. Then make a table of values.

The solution of the inequality is –1.8 ≤ x ≤ 0.82.

ANSWER

x -3 -2 -1.8 -1.5 -1 0 0.5 0.8 0.9

22 + 2x – 3 9 1 -0.1 -1.5 -3 -3 -1.5 -0.1 0.42

EXAMPLE 6 Use a quadratic inequality as a model

The number T of teams that have participated in a robot-building competition for high school students can be modeled by

Robotics

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?

EXAMPLE 6 Use a quadratic inequality as a model

T(x) > 100

7.51x2 – 16.4x + 35.0 > 100

7.51x2 – 16.4x – 65 > 0

Graph 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.

ANSWER

SOLUTION

You want to find the values of x for which:

EXAMPLE 7 Solve a quadratic inequality algebraically

Solve 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

EXAMPLE 7 Solve a quadratic inequality algebraically

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

(– 4)2 –2(– 4) = 24 >15 62 –2(6) = 24 >15


6. Robotics

Use the information in Example 6 to determine in what years at least 200 teams participated in the robot-building competition.

SOLUTION

You want to find the values of x for which:

T(x) > 200

7.51x2 – 16.4x + 35.0 > 200

7.51x2 – 16.4x – 165 > 0


Graph y = 7.51x2 – 16.4x – 165 on the domain 0 ≤ x ≤ 9.

There were more than 200 teams participating in the years 1998 – 2001.

ANSWER


7. Solve the inequality 2x2 – 7x = 4 algebraically.

SOLUTION

First, write and solve the equation obtained by replacing > with 5.

Write equation that corresponds to original inequality.

Write in standard form.

Factor.

Zero product property

2x2 – 7x = 4

2x2 – 7x – 4 = 0

(2x + 1)(x – 4) = 0

x = – 0.5 or x = 4


The numbers 4 and – 0.5 are the critical x-values of the inequality 2x2 – 7x > 4 . Plot 4 and – 0.5 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 = – 3: Test x = 2: Test x = 5:

– 3

– 4

– 2

– 1

0

1 2

3

4

5

6

7

– 5

– 6

– 7

2 (– 3)2 – 7 (– 3) > 4 2 (5)2 – 7 (3) > 4 2 (2)2 – 7 (2) > 4

The solution is x < – 0.5 or x > 4.

ANSWER

graphing quadratic inequalities

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