Steps to Solving Various Equations

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

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

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.

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.

Two Variable Equations

Find solutions Plot them on a Cartesian plane Sketch the graph

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

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,∞)