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Algebra 10: Solving Inequalities & Trigonometric Identities

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Algebra 10: Solving Inequalities & Trigonometric Identities

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Questions and Answers

What is the solution to the inequality $rac{2x-1}{3} > rac{3x+1}{2}$?

$x > -1$

$x < -1$ (correct)

$x eq -1$

$x = -1$

When solving the inequality $rac{2x-1}{3} > rac{3x+1}{2}$, what crucial step occurs when dividing by -5?

The direction of the inequality sign remains the same

The two sides are multiplied by negative numbers

The direction of the inequality sign needs to be reversed (correct)

The inequality is eliminated altogether

What is the range of the cosine function?

$R$ (all real numbers)

$[0, 1]$

$(-rac{ an(rac{ ext{pi}}{2})}{2}, rac{ an(rac{ ext{pi}}{2})}{2})$

$[-1, 1]$ (correct)

Which expression correctly represents the interval notation for the set of numbers less than or equal to -3 or greater than or equal to 3?

$(- ext{infinity}, -3] ext{ or } [3, ext{infinity})$ Signup and view all the answers

What is the period of the cosine function?

$2 ext{pi}$ Signup and view all the answers

What is the domain of the function $f(x) = (x+2)^2$?

$(- ext{infinity}, ext{infinity})$ Signup and view all the answers

Which of the following correctly describes the tangent function?

It is an odd function with a range of $R$ Signup and view all the answers

What is the important identity that relates sine and cosine?

$sin^2(x) + cos^2(x) = 1$ Signup and view all the answers

When expressing the set of numbers between -1 and 5 using interval notation, which one is correct?

$(-1, 5)$ Signup and view all the answers

In step 3 of solving the inequality $rac{2x-1}{3} > rac{3x+1}{2}$, what is the resulting expression after subtracting $9x$?

$-5x - 2 > 3$ Signup and view all the answers

Study Notes

Solving Inequalities

Various methods are used to solve inequalities, including algebraic manipulation and graphical analysis.
Key steps often involve isolating the variable, squaring both sides (with caution), and adding/subtracting constants. Example usage of these methods will vary depending on the specific type of inequality, such as linear or quadratic inequalities, or inequalities containing square roots.
Solving inequalities containing square roots requires isolating the square root. Then, both sides are squared, and the final solution is checked for extraneous solutions.

Trigonometric Identities

Reciprocal identities: sin x / cos x = tan x, 1/cos x = sec x, 1/sin x = csc x, 1/tan x = cot x
Pythagorean identities: sin² x + cos² x = 1
Other identities include: cot x = 1/tan x, sec x = 1/cos x, csc x = 1/sin x

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Description

This quiz covers methods for solving inequalities, including algebraic manipulation and graphical analysis. Additionally, it explores important trigonometric identities, such as reciprocal and Pythagorean identities essential for advanced algebra. Test your understanding of both solving techniques and identity applications.