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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Study Guide for Algebra

4.1 Simplifying rational expressions where b ≠ 0 where b ≠ 0 where b, k ≠ 0

• Factoring: 4.2 Multiplying/Dividing rational expressions where b, d ≠ 0 where b, c, d ≠ 0 where b ≠ 0 where b ≠ 0

4.4 More rational expressions/complex fractions

• Combining rational expressions:

1. Find LCD
2. Multiply to get equivalent fractions with the same denominator
3. Combine numerators
4. Reduce, if necessary

• Complex fractions:

1. Find LCD of all fractions
2. Multiply numerator and denominator by LCD
3. Simplify

4.6 Fractional equations

• Solving fractional equations:

1. Find LCD
2. Multiply both sides by LCD
3. Solve for variable

4.7 More fractional equations/Applications

• Useful equations: 5.1 Using integers as exponents • Properties: – Radicant has to be positive and as small as possible
– No fractions in the radicant
– No roots in the denominator

• You can only add/subtract fractions if the radicant is the same

• Simplify, if necessary!

• Variables: Take out as many multiples as possible

1. Isolate the root on one side
2. Square both sides
3. Solve for the variable

• If there are multiple roots, you might have to repeat the process of
isolating and squaring

5.6 Merging exponents and roots 6.1 Complex numbers the conjugate of a + bi is a − bi • Factor and solve

6.3 Completing the square • Completing the square:

1. Make sure there is no factor in front of the square term
2. Divide the coefficient in front of the x-term by 2 and square this
number
3. Add this number and immediately subtract
4. Combine and solve

• The solutions for the equation ax2 + bx + c = 0 are • Discriminant: D = b2 − 4ac

– D > 0: two distinct real solutions
– D = 0: one real solution
– D < 0: two distinct complex solutions
• the sum of the solutions is • the product of the solutions is • Which method to use:
– all coefficients small: try factoring, then quadratic formula
– b small, c large: complete the square
– all large: try factoring, complete the square

1. Make sure that the right is 0
2. Find critical points: where the left side is either 0 or undefined
3. Graph critical points on the number line
4. Pick test points and determine whether they satisfy the inequalit

8.1 Graphing parabolas  is the vertex
– k > 0: concave up
– k < 0: concave down
– |k| > 1: narrower
– |k| < 1: wider

8.2 Parabolas/Circles

• complete the square to bring parabola into standard form

• circle: where is the center and r the

8.3 Ellipses
• standard form: where A,B,C > 0 is the center of the ellipse
• extension in x-direction is • extension in y-direction is 8.4 Hyperbolas
• standard form: where A,B,C > 0 where
C > 0 and A, B have different signs is the center of the hyperbola

• finding asymptotes:
1. find the slope: 2. use point-slope with the slope and center: • determine whether it is up/down (B > 0) or left/right (A > 0)
• Determine distance of the vertices from center: or 10.1 Systems of two linear equations in two variables

• Substitution:

1. Solve one equations for one variable
2. Substitute into the OTHER equation
3. Solve, then solve for the other variable as well

• Graphing approach:

– different slopes: system consistend, one solution
– same slope, different y-intercepts: inconsistent, no solution
– same slope and y-intercept: dependent, infinitely many solutions