What does the solution $$(-\frac{1}{2}, 4]$$ represent?

The solution $$(-\frac{1}{2}, 4]$$ represents all real numbers greater than -1/2 and less than or equal to 4. The parenthesis indicates that -1/2 is not included, while the bracket indicates that 4 is included.

What is the solution to the inequality $$x^2 - 9 \\le 0$$ ?

The solution to the inequality $$x^2 - 9 \\le 0$$ is $$[-3, 3]$$ , which includes all values of x between -3 and 3, inclusive.

How do you express the solution of $$\frac{x+2}{x-4} \\ge 3$$ in interval notation?

To express the solution of $$\frac{x+2}{x-4} \\ge 3$$ in interval notation, first solve the inequality to find the critical points and test intervals. The solution is $$(4, 7]$$ .

What does the inequality $$x^2 - x - 12 > 0$$ represent graphically?

Graphically, the inequality $$x^2 - x - 12 > 0$$ represents the regions of the x-axis where the parabola defined by the equation is above the x-axis. The solution is $$(-\frac{1}{2}, 4) \\cup (4, \\infty)$$ .

Define a rational inequality.

A rational inequality is an inequality that involves a rational expression, which is a fraction where the numerator and denominator are polynomials. Examples include inequalities like $$\frac{x+1}{x-2}

What is the difference between open and closed intervals?

An open interval, denoted as $$(a, b)$$ , does not include its endpoints a and b, while a closed interval, denoted as $$[a, b]$$ , includes both endpoints. This affects the solution set of inequalities.

How do you find the endpoints of a quadratic inequality?

To find the endpoints of a quadratic inequality, set the quadratic expression equal to zero and solve for x. The solutions are the critical points that will be used to test intervals.

What is the solution to the inequality $$\frac{3x^2}{x^2+4} \\ge 1.5$$ ?

The solution to the inequality $$\frac{3x^2}{x^2+4} \\ge 1.5$$ is $$(-\frac{3}{2}, 2) \\cup (2, \\infty)$$ .

Explain the importance of understanding nonlinear inequalities in mathematics.

Understanding nonlinear inequalities is important because they appear in various real-world applications, such as optimization problems, economics, and engineering. They help in modeling situations where relationships are not linear.

What is the role of the denominator in rational inequalities?

In rational inequalities, the denominator cannot be zero, as this would make the expression undefined. Identifying values that make the denominator zero is crucial for determining critical points and valid intervals.

Nonlinear Inequalities Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

Define nonlinear inequality.

Back

2.

FLASHCARD QUESTION

Front

What is interval notation?

Back

3.

FLASHCARD QUESTION

Front

How do you solve a nonlinear inequality?

Back

To solve a nonlinear inequality, follow these steps: 1. Rearrange the inequality to one side, setting it to zero. 2. Identify critical points by solving the corresponding equation. 3. Test intervals between critical points to determine where the inequality holds true. 4. Express the solution in interval notation.

4.

FLASHCARD QUESTION

Front

Back

5.

FLASHCARD QUESTION

Front

What is the significance of critical points in solving inequalities?

Back

Critical points are values where the expression equals zero or is undefined. They are significant because they divide the number line into intervals, allowing us to test each interval to determine where the inequality holds true.

6.

FLASHCARD QUESTION

Front

Explain the concept of testing intervals in inequalities.

Back

Testing intervals involves selecting test points from each interval created by critical points to determine if the inequality is satisfied in that interval. If the test point satisfies the inequality, then all values in that interval are part of the solution.

7.

FLASHCARD QUESTION

Front

Back

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