How Do You Solve and Graph This Quadratic Inequality?

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In summary, the solution to the quadratic inequality (x^2 - 1)/(x^2 + 8x + 15) ≥ 0 is (-infinity, -5) U (1, infinity). This is determined by finding the roots of the numerator and denominator and testing the intervals on a number line, where the sign of the expression alternates. The end-points of the intervals are included in the solution due to the inequality being weak.
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mathdad
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Section 2.6
Question 50Solve the quadratic inequality.

(x^2 - 1)/(x^2 + 8x + 15) ≥ 0

x^2 - 1 = (x - 1)(x + 1)

Question:

Do I solve the numerator or denominator to find the end points?

Setting each numerator factor to 0 we get x = 1, x = -1.

Factor denominator.

x^2 + 8x + 15 = (x + 3)(x + 5)

Set each denominator factor to 0.

x + 3 = 0

x = -3

x + 5 = 0

x = -5

Question:

Do I place x = -3 & x = -5 on the number line to test each interval?

Is the correct number line as follows?

<-----(-5)-----(-3)-----(-1)-----(1)----->

It looks like we must test 5 intervals, correct?
 
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  • #2
RTCNTC said:
Section 2.6
Question 50Solve the quadratic inequality.

(x^2 - 1)/(x^2 + 8x + 15) ≥ 0

x^2 - 1 = (x - 1)(x + 1)

Question:

Do I solve the numerator or denominator to find the end points?

You want to determine the roots of both the numerator and denominator as your critical numbers (i.e., values for which the expression can change sign).

RTCNTC said:
Setting each numerator factor to 0 we get x = 1, x = -1.

Factor denominator.

x^2 + 8x + 15 = (x + 3)(x + 5)

Set each denominator factor to 0.

x + 3 = 0

x = -3

x + 5 = 0

x = -5

Question:

Do I place x = -3 & x = -5 on the number line to test each interval?

Is the correct number line as follows?

<-----(-5)-----(-3)-----(-1)-----(1)----->

Yes, that's correct.

RTCNTC said:
It looks like we must test 5 intervals, correct?

You really only need to test 1 interval, since all roots are of multiplicity 1 (odd), the sign will alternate across all intervals. Because the inequality is weak, we include the end-points of the intervals that are part of the solution. :D
 
  • #3
Is the correct number line as follows?

<-----(-5)-----(-3)-----(-1)-----(1)----->

I am going to test each interval for algebra practice.

(x - 1)(x + 1)/(x + 3)(x + 5) ≥ 0

When x = -3 & -5, we get undefined. This means we do include -3 and -5 as part of our solution. The same can be said for x = -1 & x = 1.

For (-infinity, -5), let x = -6. Here we get a true statement.

For (-5, -3), let x = -4. Here we get a false statement.

For (-3, -1), let x = -2. Here we get a false statement.

For (-1, 1), let x = 0. Here we get a false statement.

For (1, infinity), let x = 3. Here we get a true statement.

Solution: (-infinity, -5) U (1, infinity)

Correct?
 

FAQ: How Do You Solve and Graph This Quadratic Inequality?

What is a quadratic inequality?

A quadratic inequality is an inequality in the form of ax^2 + bx + c < 0, ax^2 + bx + c > 0, or ax^2 + bx + c ≤ 0, where a, b, and c are constants and x is a variable. It represents a region on a graph where the y-values are either less than or greater than the x-axis.

How do you solve a quadratic inequality?

To solve a quadratic inequality, you must first factor the quadratic expression and find its zeros. Then, you must plot these zeros on a number line and determine the intervals where the expression is positive or negative. Finally, you can use these intervals to write the solution as an inequality.

Can a quadratic inequality have more than one solution?

Yes, a quadratic inequality can have more than one solution. It can have two solutions if the parabola intersects the x-axis at two distinct points, and it can have infinitely many solutions if the parabola is entirely above or below the x-axis.

How do you graph a quadratic inequality?

To graph a quadratic inequality, you must first graph the corresponding quadratic function. Then, you can shade the region above or below the x-axis, depending on the inequality symbol. You can also use test points to determine which side of the inequality to shade.

What is the difference between a quadratic inequality and a quadratic equation?

A quadratic inequality involves an inequality symbol (<, >, ≤, or ≥) and represents a region on a graph, while a quadratic equation involves an equal sign (=) and represents the points where the parabola intersects the x-axis on a graph.

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