Chapter 13: Algebra Basics

Work through the following exercises about square roots: 6√3 -√3

5√3

4x-9=19 Solve for x

7

Work through the following exercises about square roots: √4 × √16

8

x+3=11 Solve for x

8

Rules for square roots (not a question)

Adding/Subtracting 2√5 + 4√5 = 6√5 Multiplying/Dividing 3√3 × 2√12 = 6√36 = 6(6)=36 √12/√3 = √4 = 2

7x+3y=20 3x+7y=30 Solve for x+y

5

Work through the following exercises about square roots: √60

2√15

Factor or expand the following: x²+2xy-8y²

x²+2xy-8y²= (x+4y)(x-2y)

Factor or expand the following: x²-8x+15

x²-8x+15 = (x-5)(x-3)

Express the inequality y is less than 5 and greater than or equal to 2 Variable:Y

2≤y<5

11x-y=27 3x-(2/3)y=8 Solve fore 2x+y

3

Work through the following exercises about square roots: √(9/4)

3/2

Factor or expand the following: 2y²-12y+10

2y²-12y+10= 2(y-1)(y-5)

Work through the following exercises about square roots: √4 × √10

2√10

Factor or expand the following: (3+√2)(3-√2)

(3+√2)(3-√2) = 9 -2 = 7

Factor or expand the following: (x+4)²

(x+4)² = x²+8x+16

Factor or expand the following: (y-10)²

(y-10)² = y² -20y+100

Work through the following exercises about square roots: √(5/8) + √(3/8)

(√5 +√3)/2√2

7y+4x=8 2y+x=8 Solve fore x+y

-16

3y-2x=16 5y-2x=24 Solve for x

-2

12-6x>0 Solve for x

-6x>-12→x<2

Express the inequality x is greater than 0 and less than 7 variable: x

0<x<7

Common quadratics 1) x²-y²= ? 2) (x+y)²=? 3) (x-y)²=?

1) x²-y²= (x+y)(x-y) 2) (x+y)²=(x²+2xy+y²) 3)(x-y)²= (x²-2xy+y²)

Work through the following exercises about square roots: √10 ×√10

10

(3/5)x-6=3 Solve for x

15

Work through the following exercises about square roots: 7√28 + 2√7

16√7

(y/3)-5=2 Solve for y

21

Express the inequality x is greater than 2 and y is less than x and greater than -3 Variables: x,y

2<x and -3<y<x

x is greater than -2 and less than 3 and y is greater than -3 and less than 4. Expression: xy

Compare the to inequalities by multiplying their numbers. 3(-3)=-9 3(4)=12 -2(-3)=6 -2(4)=-8 Answer: -9<xy<12

x is less than 10 and greater than 4 and y less than 6 and greater than 2 Expression: x-y

Compare the to inequalities by subtracting their numbers. For example, the biggest number in the first inequality is ten. So, write 10-2=8. or 10-6=4. Now, do the same for the smallest number. 4-6=-2 4-2=2 8 is the largest number and -2 the smallest. So... Answer: -2<x-y<8

Mark each of the following equations or inequalities as True or False (5²)⁰ = 5² × 5⁰

False

Mark each of the following equations or inequalities as True or False 2² + 2³ =2⁵

False

Mark each of the following equations or inequalities as True or False 2⁶< 4³ < 8²

False Break them down to the common denominator to see why this is the case 2⁶<(2²)³< (2³)²

3x + 10y=64 6x-10y=8 Find x and y

Fast way: add both equations to eliminate y 3x + 10y=64 + 6x-10y=8 9x=72 x=8 6(8)-10y=8 y=4 OR do the algebra 3x + 10y=64 →→ 3(4/3 +5y/3) +10y =64 and so on...

If 0≤x≤10 and -10≤y≤-1, then what is the range of x-y?

Find: x max - y max x max - y min x min - y max x min- y min 10-(-1)=11 10-(-10)=20 0-(-1)=1 0-(-10)=10 For the range of x-y the min is 1 and the max is 20. So, 1 ≤ x-y ≤ 20

Find x and y 4x+7y=41 2x+3y=19

GG easy Multiply the bottom equation x2 so you can cancel x y=3; x=5

Reminder, this is not a question. These are the rules for exponents:

Multiplication = 3²x3³=3⁵ Division= 5³÷5²= 5¹=5 Power = (6²)³= 6⁶ Negative=8³÷8⁵= 8⁻²= 1/8²= 1/64 Zero= 4³ x 4⁻³= 4⁰= 1

5x+9y=20 2x+4y=4 Find x-y

Multiply the 2nd equation by 2 and then subtract 5x+9y=20 (2x+4y=4) 5x+9y=20 - 4x+8y=8 = x-y=12 Answer is 12

Work through the following exercises about square roots: 8√3-5√5=

There is no further simplification

Mark each of the following equations or inequalities as True or False 3³ × 3⁴ < 3¹⁴

True

Mark each of the following equations or inequalities as True or False 6²/2² =3²

True 6²= 2²×3² then cancel like terms

Mark each of the following equations or inequalities as True or False 8⁸/8¹⁰ = 8⁻²

True and that equals 1/64

Work through the following exercises about square roots: √(27/9)

√3

2x-3y=8 x-2y=3 Considering the two equations above, solve for the value of x-y

x-y= 5

6x+3y=9 2x+7y=17 Find x-y

x-y=-2 Note: you can subtract the equations as they are and then divide the everything on the remaining equation by 4. This will yield x-y=-2 or, you know, you can do the old substitution method. That works too.