1

Name ______________________________ Period ________

Algebra 1

Unit 1 Practice:

Tools of Algebra:

Expressions, Equations

and Inequalities

Lesson 1: Using the Graphing Calculator to Explore Functions pg. 3

Lesson 2: Dimensional Analysis pg. 5

Lesson 3: Multiplying and Dividing Monomials pg. 6

Lesson 4: Multiplying and Dividing Polynomials pg. 7

Lesson 5: Adding and Subtracting Polynomials pg. 9

Lesson 6 & 7: The Distributive Property pg. 11

Lesson 8: The Commutative and Associative Properties pg. 13

Lesson 9: Sets of Numbers pg. 15

Lesson 10: Solving Equations Algebraically pg. 17

Lesson 11: Recognizing Properties of Equality in Solving Equations pg. 19

Lesson 12: Solving Equations with Fractions pg. 21

Lesson 13: Solving Equations for a Specific Variable pg. 23

Lesson 14: Solving Linear Inequalities Algebraically pg. 25

Lesson 15: Interpreting Two or More Inequalities Joined by “AND” or “OR” pg. 27

Lesson 16: Solving and Graphing Inequalities Joined by “AND” or “OR” pg. 29

2

3

Lesson 1: Using the Graphing Calculator to Explore Functions

Look at the GRAPH of each function first. Then use the graph to determine an

appropriate table of values to use to plot the function.

1. 𝑦 = 2𝑥 − 7

2. 𝑦 = |2𝑥 + 1| − 3

𝑥 (input) 𝑦 (output)

𝑥 (input) 𝑦 (output)

4

Complete the table of values below. Then graph the function.

3. 𝑦 = √3𝑥 + 4

4. 𝑦 = 3𝑥

𝑥 (input) 𝑦 (output)

-4

-3

-2

-1

0

4

7

𝑥 (input) 𝑦 (output)

-2

-1

0

1

2

3

4

5

Lesson2: Dimensional Analysis

Convert each measurement to the given units.

1. 261 g kg

2. 3 days seconds

3. 9,474 mm cm

4. 0.73 kL L

5. 5.93 cm3 m3

6. 498.82 cg mg

7. 1 ft3 m3 (Note: 3.28 ft = 1 m) 8. 1 year minutes

9. 175 lbs kg (Note: 2.2 lb = 1 kg) 10. 4.65 km m

6

Lesson 3: Multiplying and Dividing Monomials

Multiplying Monomials

1. xxxxx 2. yyyxx 3. 53 xx 4. yyxx 42 5. 23525 nnmnm

6. xx 52 7. zzyy 2946 8. 24 43 xx 9. 422 25 yxx 10. 23 354 xxx

Dividing Monomials

1. x

x3

2. 42

65

yx

yx 3.

3

3

x

x 4.

zyx

zyx64

6810

5. 52

623

cab

cba

6. x

x

2

2 2

7. 2

5

7

14

p

p 8.

2

6

20

10

d

d 9.

nm

nm8

108

2

12 10.

26

327

15

5

zx

zyx

Power to a Power

1. 32 )(x 2. 233 )( yx 3. 642 )( zxy 4. 35)2( x 5. 3)4( y

6. 47)6( h 7. 49)5( p 8. 24 )3( d 9. 810)( a 10. 572 )3( zyx

7

Lesson 4: Multiplying and Dividing Polynomials

Classify each as monomial, binomial, trinomial, or polynomial.

1. 2𝑥 + 1 2. 17𝑥2 + 11 3. 8𝑥3 + 2𝑥2 + 3𝑥 − 7

4. −130 5. 4𝑎2 + 7𝑎 − 10 6. 10𝑥3 − 2𝑥 + 1

One of the expressions above is referred to as a constant. Can you identify the

expression that is a constant?

Simplify each expression. Combine like terms if possible.

7. 3𝑥2(4𝑥2 + 5) 8. – 7𝑥(𝑥 − 4𝑘) 9. 9𝑥4−27𝑥6

3𝑥3

10. 𝑐(𝑐𝑚 + 𝑐𝑛) 11. (8𝑐2 − 12𝑑2) ÷ (−4) 12. 12𝑥3−6𝑥2+2𝑥

2𝑥

8

Simplify each expression. Combine like terms if possible. Leave answers in

standard form.

13. 6(𝑥2 + 2𝑥 + 7) 14. 4𝑥(1 − 𝑥) 15. −𝑥2(𝑥 + 5)

16. 3𝑥2(4𝑥3 − 5𝑥 + 10) 17. 3𝑥(−𝑥2 + 2𝑥 − 12) 18. −18𝑥2+21𝑥

−3

19. 20𝑥4−15𝑥2

5𝑥2 20. 𝑥4+3𝑥3+7𝑥

𝑥

9

Lesson 5: Adding and Subtracting Polynomials

Find each sum or difference by combining the parts that are alike.

1. (2𝑝 + 4) + 5(𝑝 − 1) − (𝑝 + 7) 2. (7𝑥4 + 9𝑥) − 2(𝑥4 + 13)

3. (5 − 𝑡2) + 6(𝑡2 − 8) − (𝑡2 + 12) 4. (8𝑥3 + 5𝑥) − 3(𝑥3 + 2)

5. (12𝑥 + 1) + 2(𝑥 − 4) − (𝑥 − 15) 6. (9 − 𝑡 − 𝑡2) −3

2(8𝑡 + 2𝑡2)

7. (4𝑚 + 6) − 12(𝑚 − 3) + (𝑚 + 2) 8. (15𝑥4 + 10𝑥) − 12(𝑥4 + 4𝑥)

10

9. The expression (𝑥2 − 5𝑥 − 2) − (−6𝑥2 − 7𝑥 − 3) is equivalent to

10. The expression (2𝑥2 + 6𝑥 + 5) − (6𝑥2 + 3𝑥 + 5) multiplied by 12

𝑥3 is

equivalent to

11. When 3𝑥2 − 8𝑥 is subtracted from 2𝑥2 + 3𝑥, the difference is

12. When 3𝑔2 − 4𝑔 + 2 is subtracted from 7𝑔2 + 5𝑔 − 1, the difference is

13. When 4𝑥2 + 7𝑥 − 5 is subtracted from 9𝑥2 − 2𝑥 + 3, the result is

11

Lesson 6 & 7: The Distributive Property

Simplify each expression using the distributive property.

1. (𝑥 + 7)(𝑥 – 5) 2. (2 – 𝑚)(4 – 𝑚) 3. (𝑏 – 4)(𝑏 + 16)

4. (3

7𝑚 +

8

3𝑛)

2 5. (𝑥 +

3

4) (𝑥 –

5

4) 6. (3𝑥 – 4)2

7. (𝑎 + 𝑐)(𝑏 + 𝑐 + 𝑎) 8. (𝑥 + 𝑦 + 𝑧)(𝑥 + 1)

9. (𝑥 + 𝑦 – 3)(𝑥 + 𝑦 + 2)

12

10. Given the expression: (𝑥 + 3)(𝑦 + 1)(𝑥 + 2)

Write an equivalent expression by applying the Distributive Property.

11. Use the distributive property to show that the equation below is true.

(𝑥 + 2𝑎)2 = 𝑥2 + 2𝑎𝑥 + 2𝑎(𝑥 + 2𝑎).

12. Use the distributive property to show that the equation below is true.

(𝑥 + 2𝑎)2 = 𝑥2 + 4𝑎(𝑥 + 2𝑎) − 4𝑎2

13

Lesson 8: The Commutative and Associative Properties Select the property that represents the equation.

1. −3 + 6 = 6 + −3

(1) Associative Property of Addition

(2) Commutative Property of Addition

2. 3 + (5 + 7) = 3 + (7 + 5)

(1) Associative Property of Addition

(2) Commutative Property of Addition

3. 7 • 3 = 3 • 7

(1) Associative Property of Multiplication

(2) Commutative Property of Multiplication

4. (7 • 5) • 2 = 7 • (5 • 2)

(1) Associative Property of Multiplication

(2) Commutative Property of Multiplication

5. Which of the following is an illustration of the associative property?

(1) 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 (3) 𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐

(2) 𝑎𝑏 + 0 = 𝑎𝑏 (4) 𝑎 + 𝑏 = 𝑏 + 𝑎

14

6. Prove (𝑥 + 𝑦) + 𝑧=(𝑧 + 𝑦) + 𝑥is true for all real numbers 𝑥, 𝑦 and 𝑧.

7. Write a mathematical proof to show that (𝑥 + 𝑎)(𝑥 + 𝑏) is equivalent to 𝑥2 + 𝑎𝑥 + 𝑏𝑥 + 𝑎𝑏.

8. The steps in finding the product of (3𝑥2𝑦5) and (7𝑥5𝑦2) are shown below. Fill in

either the associative property or the commutative property to justify each step.

(3𝑥2𝑦5)(7𝑥5𝑦2)

(3𝑥2)(𝑦4 ∙ 7)(𝑥5𝑦2) _______________________________________

(3𝑥2)(7𝑦4)(𝑥5𝑦2) _______________________________________

3(𝑥2 ∙ 7)(𝑦4𝑥5𝑦2) _______________________________________

3(7𝑥2)(𝑥5𝑦4𝑦2) _______________________________________

(3 ∙ 7)(𝑥2𝑥5)(𝑦4𝑦2) _______________________________________

21𝑥7𝑦6

15

Lesson 9: Sets of Numbers

Name all the classifications for each real number.

1. √34

2. √121

3. −3

11

4. 2. 6̅

5. 3.14

6. 𝜋

16

True or False: If the answer is false, provide an example to support your

reasoning.

7. The sum of two rational numbers is rational.

8. The product of two irrational numbers is rational.

9. The sum of a rational number and an irrational number is irrational.

10. The product of a nonzero rational number and an irrational number is irrational.

17

Lesson 10: Solving Equations Algebraically

Solve each equation. Check your solution(s).

1. 4𝑦 − 3 = 5𝑦 − 8 2. −7 − 6𝑎 + 5𝑎 = 3𝑎 − 5𝑎

3. 7 – 2𝑥 = 1 – 5𝑥 + 2𝑥 4. 4(𝑥 − 2) = 8(𝑥 − 3) − 12

5. 𝑛2 − 4n + 8 = (n − 4)(n + 3) 6. −21 − 8𝑎 = −5(𝑎 + 6)

18

7. −11 − 2𝑝 = 6𝑝 + 5(𝑝 + 3) 8. 2(6𝑏 + 8) = 4 + 6𝑏

9. 3

2𝑟=

1

4 10.

−3

1+𝑛=

−6

1−𝑛

11. A lawn-and-garden dealer wants to make a new blend of grass seed by using 200 pounds of $0.45 per pound seed and some $0.65 per pound seed. How much of the $0.65 seed does the dealer need to make a $0.55 per pound blend?

0.45(200) + 0.65(x) = 0.55(200+x)

19

Lesson 11: Recognizing Properties of Equality in Solving Equations

Solve the equation for 𝒙. For each step, describe the property used to convert

the equation.

1. 3𝑥 − [8 − 3(𝑥 − 1)] = 𝑥 + 19

2. 𝑥−3

𝑥−1=

𝑥+1

𝑥+2

20

3. Complete the Justification column with the appropriate property used to

convert the equation.

Steps: Justification

3(𝑥 − 2) + 5𝑥 = 9𝑥 − 24 Given

3𝑥 − 6 + 5𝑥 = 9𝑥 − 24 a.

3𝑥 + 5𝑥 − 6 = 9𝑥 − 24 b.

8𝑥 − 6 = 9𝑥 − 24 c.

8𝑥 − 8𝑥 − 6 = 9𝑥 − 8𝑥 − 24 d.

−6 = 𝑥 − 24 e.

−6 + 24 = 𝑥 − 24 + 24 f.

18 = 𝑥

21

Lesson 12: Solving Equations with Fractions

Solve each equation. Check your solutions.

1. 3

4𝑥 =

1

2 2.

−5

6𝑥 =

3

4

3. −1

4𝑤 − 3 = 𝑤 +

1

3 4.

1

2(5𝑥 − 2) = 2𝑥 + 4

22

5. 2

3(3𝑥 + 1) = 5 6.

1

2+

2

5𝑡 − 1 =

1

5𝑡 + 𝑡

7. 𝑥

2−

5𝑥

6=

1

9 8. 𝑦 −

2

5= −

1

3

Bonus: Can figure this one out?

2

𝑥−

3

8𝑥=

1

4

23

Lesson 13: Solving Equations for a Specific Variable

Solve each equation for 𝒙.

1. 𝑎𝑥 + 3𝑏 = 2𝑓 2. 𝑥+𝑏

4= 𝑐 3.

𝑥

5− 7 = 2𝑞

4. 𝑥

6−

𝑥

7= 𝑎𝑏 5.

3𝑎𝑥+2𝑏

𝑐= 4𝑑 6. Solve for 𝑠:

𝐴 = 𝑠2

24

Solve each equation for y. Then graph each equation on the graph provided.

Label each line with the appropriate equation.

7. −6𝑥 − 3𝑦 = 3 8. 10𝑥 − 16 = 4𝑦

9. 18 − 3𝑦 = 1𝑥 10. 6𝑦 − 4𝑥 = 30

25

Lesson 14: Solving Inequalities Algebraically

Find the solution set to each inequality. Then express the solution graphically

on the number line.

1. 2

3𝑥 ≠

1

2+ 2 2. −5(𝑥 − 1) ≥ 10 3. 13𝑥 < 9(1 − 𝑥)

4. 8𝑦 + 4 < 7𝑦 − 2 5. 6 – 𝑎 ≤ 15 6. −𝑥

12≤

14

7. 3(2𝑥+2)

6>

1

3𝑥 + 2 8. 4(𝑥 − 3) > 2(𝑥 − 2)

26

9. 8𝑥 − 6(𝑥 − 2) > 20 − 2𝑥 10. 8(𝑥 − 2) − 3(2𝑥 + 1) ≥ 7𝑥 + 4 − 3(𝑥 + 1)

11. Two siblings Edwin and Rhea are both going skiing but choose different

payment plans. Edwin’s plan charges $45 for rentals and $5.25 per lift up the

mountain. Rhea’s plan was a bundle where her entire day cost $108. (source: emathinstruction Kirk Weiler)

(a) Set up an inequality that models the number of trips, n, up the mountain for

which Edwin will pay more than Rhea. Solve the inequality.

(b) What is the greatest amount of trips that Edwin can take up the mountain

and still pay less than Rhea? Explain how you arrived at your answer.

27

Lesson 15: Interpreting Two or More Inequalities Joined by “AND” or “OR”

Graph the solution set to each compound inequality on a number line.

1. 𝒙 < −𝟖 or 𝒙 > −𝟖 2. 𝟎 < 𝒙 ≤ 𝟏𝟎 3. Graph each compound sentence on a number line.

a. 𝑥 = 2 or 𝑥 > 4 b. 𝑥 ≤ −5 or 𝑥 ≥ 2

28

4. The amount of money that Tucker carries in his wallet is at least 25 dollars but no more than 100 dollars. Create a scale on the number line below and then use it to show the amount of money that Tucker carries with him in his wallet.

29

Lesson 16: Solving and Graphing Inequalities Joined by “AND” or “OR”

Solve each inequality. Represent your solution on a number line.

1. −3𝑥 > 12 or 5𝑥 ≥ 10 2. −2𝑥 + 8 < 14 and 3𝑥 + 1 < 1

3. 3(6 − 𝑦) ≤ 6 and 6 − 𝑦 ≥ 8 4. −1 < 9 + 𝑛 < 17

30

5. 3𝑥 < 2𝑥 − 3 or 7𝑥 > 4𝑥 − 9 6. 2𝑛 + 5 > 1 and 3𝑛 + 4 > 7

7. ≤ k 8. v or v