Solving Quadratic Inequalities

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In summary, the person solved the quadratic inequality by finding the solutions in the intervals [-4, -3], [3, 4], and (3, 4).
  • #1
mathdad
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Section 2.6
Question 36Solve the quadratic inequality.

x^4 - 25x^2 + 144 ≤ 0

Can someone get me started?
 
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  • #2
RTCNTC said:
Section 2.6
Question 36Solve the quadratic inequality.

x^4 - 25x^2 + 144 ≤ 0

Can someone get me started?

Factor:

\(\displaystyle (x^2-9)(x^2-16)\le0\)

Continue factoring...:D
 
  • #3
Thank you very much. I am at the AMC about to watch the shark movie 47 Meters Down. I will answer this and the other 3 questions later this evening. Thank you again for your continual help in this website.
 
  • #4
RTCNTC said:
Thank you very much. I am at the AMC about to watch the shark movie 47 Meters Down. I will answer this and the other 3 questions later this evening. Thank you again for your continual help in this website.

I stopped going to movie theaters several years ago because all the phones lit up like flashlights in my eyes during the entire show is just too distracting for me. :D
 
  • #5
I dislike movie theatres in NYC. I rarely go to the movies because of extremely overcrowded conditions and rude people constantly moving pass me on their way to the bathroom. I really want to see this movie because I find sharks to be interesting creatures.

- - - Updated - - -
 
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  • #6
(x^2 - 9)(x^2 - 16) ≤ 0

(x - 3)(x + 3)(x - 4)(x + 4) ≤ 0

I can see right away that we must reject the follow values of x: 3, -3, 4, -4.

Our number line:

<---------(-4)------(-3)-----(3)-----(4)-------->

I will test each interval for algebra practice.

For (-infinity, -4), let x = -5. False statement for sure.

For (-4, -3), let x = -2. False statement.

For (-3, 3), let x = 0. False statement.

For (3, 4), let x = 3.5. True statement.

For (4, infinity), let x = 6. False statement.

The only solution is found in the interval (3, 4).

Correct?

Note: Most of the algebra is done on paper.
 
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  • #7
The solution is:

\(\displaystyle [-4,-3]\,\cup\,[3,4]\)

You missed an interval because -2 is not in the interval [-4,-3]... (an alarm should have gone off in your head when you saw the signs did not alternate while all roots are of odd multiplicity) and we include the end-points because the inequality is weak, and they do not cause division by zero.

The given expression is an even function, and so we should expect all behavior to be symmetrical across the $y$-axis. ;)
 
  • #8
I rushed through this problem. I can say silly errors were made but had this been a bonus question on a test, it would be a serious error. I will try to be more careful when solving each problem using the keyboard. If I had to work it out again, the silly errors would surely not be made.
 
  • #9
(x^2 - 9)(x^2 - 16) ≤ 0

(x - 3)(x + 3)(x - 4)(x + 4) ≤ 0

Setting each factor to 0, we get x = 3, -3, 4, and -4.

Our number line:

<---------(-4)------(-3)-----(3)-----(4)-------->

When x = -4, we get a true statement. The same can be said for x = -3, 3, and 4. This means they are part of the solution.

I will test each interval AGAIN for algebra practice.

For (-infinity, -4), let x = -5. False statement for sure.

For (-4, -3), let x = -3.5. True statement.

For (-3, 3), let x = 0. False statement.

For (3, 4), let x = 3.5. True statement.

For (4, infinity), let x = 6. False statement.

I understand why the solution is [-4, -3] U [3, 4].

So, any value of x in the intervals [-4, -3] and [3, 4] including -4, -3, 3, and 4 satisfy the given inequality. I may post two more quadratic inequality to get additional practice.
 

FAQ: Solving Quadratic Inequalities

1. What is a quadratic inequality?

A quadratic inequality is an inequality that contains a quadratic expression, which is an algebraic expression with at least one squared term. It typically has the form ax^2 + bx + c, where a, b, and c are constants and x is the variable. An example of a quadratic inequality is x^2 + 4x - 5 < 0.

2. How do you solve a quadratic inequality?

To solve a quadratic inequality, you first need to rearrange the inequality so that the expression is on one side and 0 is on the other side. Then, you can factor the quadratic expression and find the critical points, which are the points where the expression equals 0. Based on the sign of the expression between these critical points, you can determine the solution to the inequality. Graphing the quadratic expression can also help visualize the solution.

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

A quadratic equation is an equation that is set equal to 0 and can be solved to find the value(s) of the variable that make the equation true. A quadratic inequality, on the other hand, is an inequality that may have multiple solutions and is solved to find the range of values for the variable that make the inequality true.

4. How do you graph a quadratic inequality?

To graph a quadratic inequality, you can first graph the corresponding quadratic equation, which will create a parabola. Then, you can use a test point method to determine which region of the graph satisfies the inequality. If the test point satisfies the inequality, then the region containing the test point is part of the solution. If the test point does not satisfy the inequality, then the other region is part of the solution.

5. What real-world applications use quadratic inequalities?

Quadratic inequalities are commonly used in physics and engineering to model real-world scenarios, such as projectile motion and optimization problems. They can also be used in economics and finance to analyze profit and cost functions, as well as in computer science for designing algorithms and data structures.

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