Equations and Inequalities

Involving variables, equations and inequalities are forms of mathematical statements. While equations confirm the equality of two expressions, they are resolved by identifying variable values that validate the statement. On the other hand, inequalities denote that one expression surpasses or falls short of another. The solutions to these are the values that verify the statement, which frequently create a spectrum of options.

Solving for a Variable

Solve for x in the equation

5 x - 2 = 3 x + 4

.

Converting from Interval to Inequality

Convert the interval notation

(- \infty, 3]

to inequality notation.

Solve by Completing the Square

Solve the equation by completing the square:

x^{2} - 6 x + 5 = 0

Finding the Domain

Find the domain of the function

f (x) = \frac{1}{\sqrt{5 - x}}

Finding the Range

Find the range of the function

y = 2 x^{2} - 3 x + 1

when

x

is in the interval

[- 2, 3]

.

Finding the Domain and Range

Find the domain and range of the function

f (x) = \sqrt{4 - x}

Finding the Asymptotes

Find the vertical and horizontal asymptotes of the function

y = \frac{2 x^{2} - 3 x + 1}{x - 1}

.

Solving by Factoring

Solve the following equation by factoring:

x^{2} - 5 x + 6 = 0

Solving Rational Equations

Solve the rational equation

\frac{x}{x - 3} - \frac{2}{x + 1} = \frac{1}{x^{2} - 2 x - 3}

.

Quadratic Formula

Find the roots of the quadratic equation

3 x^{2} + 7 x - 6 = 0

using the quadratic formula.

Quadratic Inequalities

Solve the quadratic inequality:

x^{2} - 4 x - 5 \leq 0

.

Rational Inequalities

Solve the following rational inequality:

- \frac{2}{x - 3} \geq 1

Finding the Discriminant

Find the discriminant of the quadratic equation

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

Finding the Quadratic Constant of Variation

Given the quadratic function

y = a x^{2} + b x + c

passes through the points (1,7), (2,11), (3,17), determine the constant of variation

a

.

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