College Algebra : Polynomial Inequalities

Study concepts, example questions & explanations for College Algebra

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Example Questions

Example Question #1 : Polynomial Inequalities

Solve 

Possible Answers:

Correct answer:

Explanation:

First we substract  on each side

 

Now we can factor the left hand side

 

From this, we can see that the roots are at , and .

To make this inequality true, we will have two answers.

 

 

Example Question #2 : Polynomial Inequalities

When is the inequality  true?

Possible Answers:

this inequality cannot be satisfied.

Correct answer:

Explanation:

We can rewrite this as . In order to solve the inequality we need to factor the polynomial. Taking a couple educated guesses we find that this polynomial factors as . So now we have our inequality written as

.

When is this true? When both linear factors are positive, and also when they are both negative. 

If we match them up then we do get that the inequality is satisfied when  and when .

Example Question #3 : Polynomial Inequalities

Give the solution set of the inequality

Possible Answers:

The set of all real numbers

Correct answer:

Explanation:

The boundaries of the intervals in the solution set of the polynomial inequality 

are the zeroes of the equation

By the Zero Product Property, either

,

in which case

,

or 

,

in which case

 and 4 are the boundaries of the intervals, which are , and . To determine which intervals belong to the solution set, choose one value from each, substitute for  in the original inequality, and determine whether it is true.

: Choose 

False - do not include 

 

: Choose 

True - include 

 

: Choose 

True - include 

 

Also, the boundary values themselves should not be included, because the inequality symbol  does not allow for equality.

 

The solution set is therefore 

Example Question #4 : Polynomial Inequalities

Give the solution set of the inequality

Possible Answers:

Correct answer:

Explanation:

The boundaries of the intervals in the solution set of the polynomial inequality 

are the zeroes of the equation

By the Zero Product Property, either

,

in which case

,

or 

,

in which case

 and 5 are the boundaries of the intervals, which are , and . To determine which intervals belong to the solution set, choose one value from each, substitute for  in the original inequality, and determine whether it is true.

: Choose 

False - do not include 

 

: Choose 

False - do not include 

 

: Choose 

True - include 

 

Also, the boundary values themselves should be included, because the inequality symbol  allows for equality.

 

The solution set is therefore .

Example Question #5 : Polynomial Inequalities

Give the solution set of the inequality

Possible Answers:

Correct answer:

Explanation:

To solve the polynomial inequality

it is necessary to get it into standard form by writing all terms at the same side. Subtract  from both sides:

The values of  that serve as boundaries of the solutions set are the zeroes of the polynomial, so solve for  in the equation:

Factor the polynomial by grouping and taking out the GCFs:

The polynomial cannot be factored further, as , as the sum of squares, is prime. By the Zero Factor Property, one of the binomials is equal to 0, so

in which case

or 

,

which has no real solution.

Therefore, the only boundary of the intervals is 7, which divides the real numbers into two intervals,  and . To determine which intervals belong to the solution set, choose one value from each, substitute for  in the original inequality, and determine whether it is true.

: Choose 

False - do not include 

 

: Choose 

True - include 

Also, since the inequality symbol  does not allow for equality, 7 is not included in the solution. The solution set is .

Example Question #6 : Polynomial Inequalities

Give the solution set of the inequality

Possible Answers:

Correct answer:

Explanation:

To solve the polynomial inequality

it is necessary to get it into standard form by writing all terms at the same side. Subtract  from both sides:

The values of  that serve as boundaries of the solutions set are the zeroes of the polynomial, so solve for  in the equation:

Factor the polynomial by grouping and taking out the GCFs:

 is the difference of squares and can be factored according to a pattern:

By the Zero Factor Property, one of the binomials is equal to 0, so

in which case

,

in which case

,

or

in which case

.

, and 2 are the boundaries of the intervals, which are , and . To determine which intervals belong to the solution set, choose one value from each, substitute for  in the original inequality, and determine whether it is true.

: Choose 

True - include 

 

: Choose 

False - do not include 

 

: Choose 

True - include 

 

: Choose 

False - do not include 

 

Also, the boundary values themselves should not be included, because the inequality symbol  does not allow for equality.

The solution set is 

Example Question #7 : Polynomial Inequalities

Which is correct graph of the following function?

Possible Answers:

None of the other graphs.

Wrong1

Wrong3

Solutions

Wrong2

Correct answer:

Solutions

Explanation:

First graph 

Since the inequality is , this means that the line is included in the solution, as indicated by a solid line.

Shade the area outside of the parabola since it is a negative parabola, but greater than or equal to.

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