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Inequalities

100

x + 5 >/ 8

x >/ 3

Work: 

Subtract 5 from both sides

200

10x + 5 < 11 - 2x

Solve for x.

x <1/2

Work: 

10x + 5 < 11 - 2x

      - 5     - 5

12x < 6 (Divide both sides by 12)

x < 1/2

300

Solve.

(8^2)x + 1073 < 92x +625

x > 16

Work:

(8^2)x + 1073 < 92x +625

(Start with 8^2 then subtract 625 from each side)

64x + 448 < 92x (subtract 64x from both sides)

448 < 28x (divide both sides by 28)

16 < x

400

-3 (3x + 6) >/ 7 (x - 2)

Solve.

-1/4 >/ x

-3 (3x + 6) >/ 7 (x - 2) (distribute -3 and 7 on both sides)

-9x -18 >/ 7x -14 (add 14) 

-9x -4 >/ 7x (add 9x) 

-4 >/ 16x (divide both sides by 16)

-1/4 >/ x

500

1- 4x </ 21 and 5x + (7^2) >/ 69

Solve.

x >/ -5  and x >/ 4

1- 4x </ 21 (subtract 1) 

-4x </ 20 (divide by -4 & flip the sign)

x >/ -5

5x + (7^2) >/ 69 (solve 7^2)

5x + 49 >/ 69 (subtract 49)

5x >/ 20 (divide by 5)

x >/ 4

Equation Forms

100

What is slope intercept form?

y = mx + b

200

What is standard form?

Ax + By = C

300

What is point slope form?

y - y₁ = m (x - x₁)

400

y + 7 = 14(x+4)

Solve for y

y=14x + 49

Work:  y + 7 = 14(x+4) 

(Distribute the 14 to inside the parenthesis)

y + 7 = 14x + 56 (subtract 7 from both sides)

y=14x + 49

500

Solve: -3(4x+3)+4(6x+1)=43

x=4 

Work:

-3(4x+3)+4(6x+1)=43 (substitute)

-12x-9+24x+4=43 (combine like terms)

12x-5=43 (add 5)

12x=48 (divide 12)

x=4

Solving

100

   2y + 7x = -5

+ 5y -7x = 12

Solve for y using elimination

y = 1

Work: 

   2y + 7x = -5

+ 5y -7x = 12    

After adding you get: 7y + 0 = 7 (divide both sides by 7)

y = 1

200

2x+5y=19

y=3x -3

Solve for x and y using substitution 

x=2 , y=3

2x+5(3x-3)=19 (substitute)

2x+15x-15=19 (add like terms) 

17x=34 (divide 12)

x=2

y=3(2)-3 

y=6-3

y=3

300

-2x + y = 4

-2x + 3y = 4

Solve for x and y using elimination

y = 1 and x = -1

Work: 

-2x + y = 4

-2x + 3y = 4 (add the two equations)

4y = 8 (divide both sides by 4)

Plug in y = 2 to either equation (we did the -2x + y = 4)

-2x + (2) = 4 (subtract 2 on both sides)

-2x = 2 (divide both sides by -2)

x = -1

400

3x + 4y = 12

4x - 8y = -12

Solve for x and y by elimination

x = 1.2 and y = 2.1

Work: Multiply both sides of 3x + 4y = 12 to get -

6x + 8y = 24 

4x - 8y = -12 (add)

10x = 12 (divide both sides by 10)

x = 1.2 (plug in to either equation, we did  4x - 8y = -12)

4 (1.2) - 8y = -12

4.8 -8y = -12 (subtract 4.8 from both sides)

-8y = 16.8 (divide both sides by 8)

y = 2.1

500

5r + 2s = 23

4r + s = 19

Solve with substitution

r = 2

s = -1

Work: To get "s" alone subtract 4r from both sides (4r + s = 19)

Then you get s = -4r + 19 (plug in to 5r + 2s = 23) 

5r + 2(-4r + 19) = 23

5r - 8r + 38 = 23 (Combine like terms then subtract 38 from both sides)

-3r= -15

r = 2

Binomials

100

How many terms does a binomial have?

2 terms

200

How many terms does a polynomial have?

2 or more terms (2 = binomial)

300

(9x-4)(x-6) 

Solve for x

x = 4/9, 6

400

x² + 7x - 18

Solve for x

x=2,−9

Work: x² + 7x - 18 = 0 (add 18 to both sides)

x² + 7x = 18 (factor)

(x + 2) (x - 9)

x =2, -9

500

What is a polynomial called with 4 terms?

A quadrinomial

Exponents

100

Simplify: (x³)⁴

x¹² 

You multiply the exponents outside the parenthesis with the ones inside.

200

Simplify: (5x³) (4x⁷)

20x¹⁰

Work: (5x³) (4x⁷) (Multiply 5 and 4)

20x³x⁷ (Combine like terms: x)

20x¹⁰

300

Simplify: (-2x³y⁴)²

4x⁶y⁸

Work: 

Distribute the 2 to everything in the parenthesis

400

Simplify: (5x³y⁴z⁵)³

125x⁹y¹²z¹⁵

Work: 

(Distribute the 3 to everything in the parenthesis)