Algebra

equations & inequalities (one variable, linear)

100

3(q − 7) = 27

q = 16

200

16 − 2t = t + 9 + 4t

t = 1

300

−67b + 6 ≤ 9b + 43

b ≥ −37/76 

400

11q + 5 ≤ 49

q ≤ 4 

500

−9x + 5 < 17  AND  13x + 25 < −1

No Solutions

System of equations

100

−9x − 6y = 15

9x − 10y = 145

x = 5 

y = -10

200

6x − 2y = 8

4x − 3y = 2

x = 2 , y = 2 

300

4x + 3y = 19

−x + 4y = 0

x = 4 , y = 1

400

6x + y = 15

−7x − 2y = −10

x = 4 , y = −9

500

x + 2y + 5z = -17

2x − 3y + 2z = -16

3x + y − z = 3 

x = -1

y = 2

z = -4

Two-variable inequalities

100

Is (-4 , -4) a solution to the inequality  y > −6x − 1

No

200

Is (1 , -2) a solution to the system of inequalities

9x + 4y < 8

−3x − 7y ≥ 5

Yes

300

Is (-4 , 2) a solution to the inequality

4x + 5y ≤ −7

No

400

Horace is a professional hair stylist.  Let C represent the number of child haircuts and A represent the number of adult haircuts that Horace can give within 7 hours.  0.75C + 1.25A ≤ 7

Horace gave 5 child haircuts. How many adult haircuts at most can he give with the remaining time?

At most 2 adult haircuts

500

An electric filter purifies air at the rate of 35 liters per minute and uses energy at the rate of 0.8, point, 8 Joules per minute. It also purifies water at the rate of 25 liters per minute and uses energy at the rate of 1.3 Joules per minute.  The filter is expected to purify more than 1000 liters of air and water while using less than 170 Joules of energy. Let A denote the number of minutes it spends purifying air and W the number of minutes it spends purifying water. Write a System of inequalities to represent this . 

35A + 25W > 1000

0.8A + 1.3W < 170

Quadratic systems

100

y = 2x + 1

(x + 1)^2 + y^2 = 1

Find both solutions

(-1 , -1)

(-0.2 , -0.6)

200

x − y = −4

y = 5(x + 1)^2 − 3

(-2 , 2)

(0.2 , 4.2)

300

2x + y = −4

y = (x + 1)^2 − 2

(-3 , 2)

(-1 , -2)

400

x^2 + y^2 = 1

y = 2x + 2

(-1 , 0)

(-0.6 , 0.8)

500

(x − 1)^2 + (y − 1)^2 = 1

y = 2x + 1

(0 , 1)

(0.4 , 1.8)

Rational expressions, equations, & functions

100

4x − 32

x^2 − 64

Reduce the Rational expression to lowest terms 

4 / x + 8

200

2x^2 + 4x

x^2 + 5x + 6

2x / x + 3

300

x − 3 / x + 1 = 4x − 6 / (x + 1)(x + 2)

Solve the Rational equation

x = 0 , 5

400

x + 1 / x − 3 = −x^2 − 6x + 1 / (x − 3)(x − 1)

x = 0

500

f(x) = 6x^3 − x^2 + 7 / 2x + 5

f(x) = ∞ ,x = -∞

f(x) = ∞ ,x = ∞