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Depdendent Variable

Number of equations to solve:

Equ. #1:

Equ. #2:

Equ. #3:

Equ. #4:

Equ. #5:

Equ. #6:

Equ. #7:

Equ. #8:

Equ. #9:

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Enter inequality to solve, e.g. 2x+3>4

Enter inequality to graph, e.g. y<3x^2-1

Dependent Variable

Number of inequalities to solve:

Ineq. #1:

Ineq. #2:

Ineq. #3:

Ineq. #4:

Ineq. #5:

Ineq. #6:

Ineq. #7:

Ineq. #8:

Ineq. #9:

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Facts to Remember

1. Properites of Inequalities:

(a) If x < y and y < z then x < z.

(b) If x < y then x +z < y + z.

(c) If x < y and z is positive, then x z < y z.
However, if x < y and z is negative, then x z > y z.

(d) |x| < y if and only if −y < x < y.

(e) |x| > y if and only if either x > y or x < −y.

2. Properties of Exponents: Be aware that there are some natural assumptions a

In particular,
x⁰ = 1 and
.
(c)

(d) These are obtained by combining the above rules:

and

3. The determinant of a 2 by 2 matrix:

4. Cramer’s Rule: To solve the system of equations

first calculate:

and

If Δ ≠ 0 then the system has a unique solution

and

If Δ = 0 and one of Δ_x, Δ_y is non zero, then the system is inconsistent i.e. has no
solution!

If Δ = Δ_x = Δ_y = 0, then the equations are essentially the same and have infinitely
many solutions, provided at least one term with the variables is present.

5. The distance between two points A and B on the real line is d (A,B) = |A − B|.

6. The distance between two points and in the xy-plane: is

7. For two points and , the midpoint is
This evaluates to:

8. For the line containing two points and , a parametric two point
form is

9. For the line containing two points and , a compact parametric
two point form is

10. For the line containing two points and , the two point form is

11. For the line containing two points and , the slope is

Further, the slope intercept form of the line is

y = m x + c,

where m is the slope and c is the y-intercept given by

12. If p is the x-intercept and q is the y-intercept of a line, then the intercept form
of the line is

13. The equation of a circle with center at (a, b) and of radius r is

14. A parametric form of a circle centered at the origin and of radius r:

where m is the parameter.

15. The quadratic formula for solutions to ax² + bx + c = 0, when a ≠ 0, is

16. Trigonometric Functions:

17. Trigonometric Identites:

Fundamental Identity.
Variant 1.
Variant 2.

Basic Identity.

That is, Sine is odd.

That is, Cosine is even.

Addition rule for sine.

Addition rule for Cosine.

Double angle formula for Sine.

Double angle formula for Cosine.

Variant 1.

Variant 2.

18. Derivative Formulas:

(a) If f(x) = p, where p is a constant, then f' (x) = 0.

(b) Power rule If , then

(c) If c is a constant and g (x) = c f (x) then g' (x) = c f' (x).