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College Algebra : Polynomial Inequalities

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Example Questions

Example Question #1 : Polynomial Inequalities

When is the inequality x^2-2x > 63 true?

Possible Answers:

-7 < x < 9

$x < -7 \; ; \; x > 9$

this inequality cannot be satisfied.

x > 8

Correct answer:

$x < -7 \; ; \; x > 9$

Explanation:

We can rewrite this as x^2-2x-63 > 0 . In order to solve the inequality we need to factor the polynomial. Taking a couple educated guesses we find that this polynomial factors as (x+7)(x-9) . So now we have our inequality written as

(x+7)(x-9) > 0 .

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

$x-9 > 0 \rightarrow x > 9$

$x-9 < 0 \rightarrow x < 9$

$x+7 > 0 \rightarrow x > -7$

$x+7 < 0 \rightarrow x < -7$

If we match them up then we do get that the inequality is satisfied when x < -7 and when x > 9 .

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Example Question #1 : Polynomial Inequalities

Give the solution set of the inequality

$(x- 4)^{2} (x+ 4 )>0$

Possible Answers:

$\left \{ x | -4 < x < 4 \textrm{ or }x> 4 \ \right \}$

The set of all real numbers

$\left \{ x | x> -4 \ \right \}$

$\left \{ x | x< -4\textrm{ or } -4 < x < 4 \ \right \}$

$\left \{ x | -4 < x < 4 \ \right \}$

Correct answer:

$\left \{ x | -4 < x < 4 \textrm{ or }x> 4 \ \right \}$

Explanation:

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

$(x- 4)^{2} (x+ 4 )>0$

are the zeroes of the equation

$(x- 4)^{2} (x+ 4 )= 0$

By the Zero Product Property, either

x-4 = 0 ,

in which case

x = 4 ,

x+ 4 = 0 ,

in which case

x = -4

and 4 are the boundaries of the intervals, which are $(-\infty , -4)$ , (-4, 4) , and $(4, \infty)$ . 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.

$(-\infty , -4)$ : Choose x = -5

$(x- 4)^{2} (x+ 4 )> 0$

$(-5- 4)^{2} (-5+ 4 )> 0$

$(-9)^{2} (-1 )>0$

81 (-1 )> 0

-81 > 0

False - do not include $(-\infty , -4)$

(-4, 4) : Choose x = 0

$(x- 4)^{2} (x+ 4 )> 0$

$(0- 4)^{2} (0+ 4 )>0$

$( - 4)^{2} (4) > 0$

16(4)>0

64> 0

True - include (-4, 4)

$(4, \infty)$ : Choose x = 5

$(x- 4)^{2} (x+ 4 )> 0$

$(5- 4)^{2} (5+ 4 )>0$

$1^{2}(9)>0$

1(9)>0

9 > 0

True - include $(4, \infty)$

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

The solution set is therefore $\left \{ x | -4 < x < 4 \textrm{ or }x> 4 \ \right \}$

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Example Question #1 : Polynomial Inequalities

Give the solution set of the inequality

$(x- 5) (x+ 5 )^{2} \ge 0$

Possible Answers:

$\left \{ x | x \ge 5 \right \}$

$\left \{ x | x = -5 \textrm{ or } x \ge 5 \right \}$

$\left \{ x | -5 \le x \le 5 \right \}$

$\left \{ x | x \le - 5 \right \}$

$\left \{ x | x \le -5 \textrm{ or } -5 \le x \le 5 \right \}$

Correct answer:

$\left \{ x | x = -5 \textrm{ or } x \ge 5 \right \}$

Explanation:

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

$(x- 5) (x+ 5 )^{2} \ge 0$

are the zeroes of the equation

$(x- 5) (x+ 5 )^{2} = 0$

By the Zero Product Property, either

x-5 = 0 ,

in which case

x = 5 ,

x+ 5 = 0 ,

in which case

x = -5

and 5 are the boundaries of the intervals, which are $(-\infty , -5)$ , (-5,5) , and $(5, \infty)$ . 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.

$(-\infty , -5)$ : Choose x = -6

$(x- 5) (x+ 5 )^{2} \ge 0$

$(-6- 5) (-6+ 5 )^{2} \ge 0$

$(-11) (-1 )^{2} \ge 0$

$(-11) (1 ) \ge 0$

$-11 \ge 0$

False - do not include $(-\infty , -5)$

(-5,5) : Choose x = 0

$(x- 5) (x+ 5 )^{2} \ge 0$

$(0- 5) (0+ 5 )^{2} \ge 0$

$( - 5) ( 5 )^{2} \ge 0$

$( - 5) (2 5 ) \ge 0$

$-125 \ge 0$

False - do not include (-5,5)

$(5, \infty)$ : Choose x = 6

$(x- 5) (x+ 5 )^{2} \ge 0$

$(6- 5) (6+ 5 )^{2} \ge 0$

$(1) (11 )^{2} \ge 0$

$(1) (121 ) \ge 0$

$121 \ge 0$

True - include $(5, \infty)$

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

The solution set is therefore $\left \{ x | x = -5 \textrm{ or } x \ge 5 \right \}$ .

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Example Question #2 : Polynomial Inequalities

Give the solution set of the inequality

$x^{3} + 4x >7x^{2} + 28$

Possible Answers:

$\left \{ x | x < 7\right \}$

$\left \{ x | x< -2 \textrm{ or } 2 < x < 7\right \}$

$\left \{ x | x > 7\right \}$

$\left \{ x | x< -2 \textrm{ or } x > 7\right \}$

$\left \{ x | -2 < x< 2 \textrm{ or } x > 7\right \}$

Correct answer:

$\left \{ x | x > 7\right \}$

Explanation:

To solve the polynomial inequality

$x^{3} + 4x >7x^{2} + 28$

it is necessary to get it into standard form by writing all terms at the same side. Subtract $7x^{2} + 28$ from both sides:

$x^{3} + 4x - (7x^{2} + 28 ) >7x^{2} + 28 - (7x^{2} + 28 )$

$x^{3}- 7x^{2} + 4x -28 >0$

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

$x^{3}- 7x^{2} + 4x -28 =0$

Factor the polynomial by grouping and taking out the GCFs:

$(x^{3}- 7x^{2} )+( 4x -28) =0$

$x^{2}(x - 7 ) + 4 (x - 7 )=0$

$(x^{2} + 4 )(x - 7 )=0$

The polynomial cannot be factored further, as $x^{2}+ 4$ , as the sum of squares, is prime. By the Zero Factor Property, one of the binomials is equal to 0, so

x-7 = 0

in which case

x = 7

$x^{2}+ 4 = 0$

$x^{2}+ 4 - 4 = 0 - 4$

$x^{2} = -4$ ,

which has no real solution.

Therefore, the only boundary of the intervals is 7, which divides the real numbers into two intervals, $(-\infty , 7)$ and $(7, \infty)$ . 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.

$(-\infty , 7)$ : Choose x = 0

$x^{3} + 4x >7x^{2} + 28$

$0 ^{3} + 4 (0) >7 \cdot 0^{2} + 28$

0+0 >0+ 28

0 > 28

False - do not include $(-\infty , 7)$

$(7, \infty)$ : Choose x= 8

$x^{3} + 4x >7x^{2} + 28$

$8^{3} + 4 \cdot 8 >7 \cdot 8^{2} + 28$

$512+ 32 >7 \cdot 64 + 28$

512+ 32 >448 + 28

544 >476

True - include $(7, \infty)$

Also, since the inequality symbol does not allow for equality, 7 is not included in the solution. The solution set is $\left \{ x | x > 7\right \}$ .

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Example Question #1 : Polynomial Inequalities

Give the solution set of the inequality

$x^{3} + 7x^{2} < 4x+ 28$

Possible Answers:

$\left \{ x| x < -7 \textrm{ or } -2 < x < 2 \right \}$

$\left \{ x| x < -7 \right \}$

$\left \{ x| x > -7 \right \}$

$\left \{ x| -2 < x < 2 \right \}$

$\left \{ x| -7 < x < -2 \textrm{ or } x > 2 \right \}$

Correct answer:

$\left \{ x| x < -7 \textrm{ or } -2 < x < 2 \right \}$

Explanation:

To solve the polynomial inequality

$x^{3} + 7x^{2} < 4x+ 28$

it is necessary to get it into standard form by writing all terms at the same side. Subtract 4x+ 28 from both sides:

$x^{3} + 7x^{2} - (4x+ 28) < 4x+ 28 - (4x+ 28)$

$x^{3} + 7x^{2} - 4x-28 < 0$

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

$x^{3} + 7x^{2} - 4x-28 = 0$

Factor the polynomial by grouping and taking out the GCFs:

$(x^{3} + 7x^{2}) - (4x+28) = 0$

$x^{2} (x + 7) - 4(x + 7) = 0$

$(x^{2} - 4)(x + 7) = 0$

$x^{2} - 4$ is the difference of squares and can be factored according to a pattern:

$(x^{2} - 2^{2})(x + 7) = 0$

(x+2)(x-2)(x+7) = 0

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

x-2 = 0

in which case

x = 2 ,

x+2 = 0

in which case

x = -2 ,

x+7 = 0

in which case

x = -7 .

, , and 2 are the boundaries of the intervals, which are $(-\infty, -7)$ , (-7, -2) , (-2,2) , and $(2, \infty)$ . 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.

$(-\infty, -7)$ : Choose x = -8

$x^{3} + 7x^{2} < 4x+ 28$

$(-8)^{3} + 7 \cdot (-8)^{2} < 4(-8)+ 28$

$-512 + 7 \cdot 64 < -32+ 28$

-512 + 448 < -32+ 28

-64 < -4

True - include $(-\infty, -7)$

(-7, -2) : Choose x = -3

$x^{3} + 7x^{2} < 4x+ 28$

$(-3)^{3} + 7 \cdot (-3)^{2} < 4(-3)+ 28$

$-27 + 7 \cdot 9 < -12 + 28$

-27 + 63 < -12 + 28

36 < 16

False - do not include (-7, -2)

(-2,2) : Choose x= 0

$x^{3} + 7x^{2} < 4x+ 28$

$0^{3} + 7 \cdot 0^{2} < 4 \cdot 0 + 28$

0+0 < 0 + 28

0 < 28

True - include (-2,2)

$(2, \infty)$ : Choose x = 3

$x^{3} + 7x^{2} < 4x+ 28$

$3^{3} + 7 \cdot 3^{2} < 4 \cdot 3+ 28$

$27 + 7 \cdot 9 < 12+ 28$

27 + 63 < 12+ 28

90 < 40

False - do not include $(2, \infty)$

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

The solution set is $\left \{ x| x < -7 \textrm{ or } -2 < x < 2 \right \}$

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Example Question #6 : Polynomial Inequalities

Which is correct graph of the following function?

$y\geq-3x^{2}$

Possible Answers:

Wrong2

Wrong1

Wrong3

None of the other graphs.

Solutions

Correct answer:

Solutions

Explanation:

First graph $y=-3x^{2}$

Since the inequality is $\geq$ , 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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Example Question #4 : Polynomial Inequalities

Solve

x^2-10>3x

Possible Answers:

$0<x<\infty$

$-\infty<x<-2$

$-\infty<x<\infty$

$5<x<\infty$

$-\infty<x<-2$

$5<x<\infty$

Correct answer:

$5<x<\infty$

$-\infty<x<-2$

Explanation:

First we substract on each side

x^2-10>3x

x^2-3x-10>0

Now we can factor the left hand side

x^2-3x-10>0

(x-5)(x+2)>0

From this, we can see that the roots are at x=5 , and x=-2 .

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

$5<x<\infty$

$-\infty<x<-2$

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College Algebra : Polynomial Inequalities

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