Solve the inequality: $$x^2 - 5x + 4 \\geq 0$$ . What are the critical points?

The critical points are found by solving $$x^2 - 5x + 4 = 0$$ , which gives $$x = 1$$ and $$x = 4$$ .

What does the solution set $$(-\infty, 1] \\cup [4, \infty)$$ represent?

It represents all values of $$x$$ that are less than or equal to 1 or greater than or equal to 4.

Solve the inequality: $$x^3 > 9x^2$$ . What is the first step?

Factor the inequality to get $$x^2(x - 9) > 0$$ .

What are the critical points for the inequality $$x^3 > 9x^2$$ ?

The critical points are $$x = 0$$ and $$x = 9$$ .

What does the solution set $$(-\infty, 0) \\cup (9, \infty)$$ indicate?

It indicates that the inequality holds true for values of $$x$$ less than 0 and greater than 9.

Solve the inequality: $$(x + 1)(x^2 + x + 1) > 0$$ . What is the first step?

Identify the roots of the factors. The factor $$x + 1 = 0$$ gives $$x = -1$$ , while $$x^2 + x + 1$$ has no real roots.

What is the solution set for the inequality $$(x + 1)(x^2 + x + 1) > 0$$ ?

The solution set is $$(-\infty, -1) \\cup (1, \infty)$$ .

Solve the inequality: $$10 - x^2 \geq 0$$ . What are the critical points?

The critical points are found by solving $$10 - x^2 = 0$$ , which gives $$x = \pm \sqrt{10}$$ .

What does the solution set $$[-\sqrt{10}, \sqrt{10}]$$ represent?

It represents all values of $$x$$ between $$-\sqrt{10}$$ and $$\sqrt{10}$$ , inclusive.

Solve the inequality: $$x^2 - 3x - 4

The critical points are found by solving $$x^2 - 3x - 4 = 0$$ , which gives $$x = -1$$ and $$x = 4$$ .

What does the solution set $$(-1, 4)$$ indicate?

It indicates that the inequality holds true for values of $$x$$ between -1 and 4, not including -1 and 4.

What is the significance of testing intervals in polynomial inequalities?

Testing intervals helps determine where the polynomial is positive or negative, which is essential for solving inequalities.

Define a critical point in the context of polynomial inequalities.

A critical point is a value of $$x$$ where the polynomial equals zero or is undefined, which helps in determining the intervals for testing the inequality.

MAT 171 Unit 4 Review 2 Polynomial Inequalities Flashcard Flashcards

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1.

FLASHCARD QUESTION

Front

Define a polynomial inequality.

Back

2.

FLASHCARD QUESTION

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What is the general method for solving polynomial inequalities?

Back

1. Solve the corresponding polynomial equation to find critical points. 2. Use these points to divide the number line into intervals. 3. Test each interval to determine where the inequality holds true.

3.

FLASHCARD QUESTION

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4.

FLASHCARD QUESTION

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5.

FLASHCARD QUESTION

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6.

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