Steps to Solving Various Equations

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Quadratic Inequalities

Move everything to one side

Factor

Find critical numbers

Plot critical numbers on number line

Test each interval

Write sets in interval notation

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Rational Inequalities

Move everything to one side

Create one fraction

Set numerators and denominators equal to 0

Plot critical numbers on number line

Test each interval

Write each set in interval notation

Always use parentheses for denominator critical numbers

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Absolute Value Equations

Isolate absolute value.

Check that the constant > 0.

If the constant is a (-), the answer is NO SOLUTION or Ø.

If the constant is 0, only use one equation.

Solve for + and - of the quantity.

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Absolute Inequalities

Isolate absolute value.

Check that the constant > 0.

If the constant is a (-) and symbol is < or ≤, the answer is NO SOLUTION or Ø.

If the constant is a (-) and symbol is > or ≥, the answer is (-∞,∞).

If the constant is 0, the answer is the critical number.

If the symbol is a > or ≥, write two equations (x>k, x<-k)

If the symbol is a < or ≤, write one compound inequality.

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Two Variable Equations

Find solutions

Plot them on a Cartesian plane

Sketch the graph

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Circle formula

(x-h)²+(y-k)²=r² ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ ​​​​​​​ center-radius form

(h,k) is the center point

The radius is r

If r²>0, the graph is a circle

If r²=0, the graph is a point

If r²<0, the graph is nonexistent

x²+y²+Dx+Ey+F=0 (where D,E,F are real) general form

Convert to center-radius form by completing the square

(x+ D/2)² + (y+ E/2)² = -F + (D/2)² + (E/2)²

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Function's Domain

Solve for y and look for ±

If there is a ±, it is not a function

If there is no ±, it is a function

POLYNOMIAL = Always (-∞,∞)

RADICAL = [radicand, ∞)

RATIONAL = set denominator equal to zero, (-∞,k) U (k,∞)