Algebra/Chapter 3/Exercises

A set of exercises related to concepts from Chapter 3.

This set contains 26 exercises (7 Conceptual Questions + 19 Exercises + 0 Projects)

Conceptual Questions

Q3.1 (Explaining Equations) In no more than one paragraph, explain in your own words what an "equation" is. In your definition, discuss an equation's purpose, as well as what a solution of an equation means.

Q3.2 (Anatomy of an Equation) Given the statement x + 6 = 12, answer the following.

1. Is this statement true or false?
2. Does x=5 make the statement true?
3. Does x=6 make the statement true?

Q3.3 (Identifying Equations) Determine which of the following are equations.

1. x + 4
2. a - 64 = 45
3. 7 < 8
4. a + 6 > 9
5. 1 + 7 = 8

Q3.4 (Special Solution Sets) Make up an equation whose solution is the null set, and explain why this is the solution set. After this, make up an equation whose solution is the set of all real numbers, and explain why this is the solution set.

Q3.5 (Equivalent Equations) Are the statements 9x + 5 = 4 and 4 = 9x + 5 the same equation? Explain your reasoning.

Q3.6 (Additive and Multiplicative Inverses) In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.

Q3.7 (Inverse Operations) What is the inverse operation of “I put my shoes on today, and I walk out of my house”?

Exercises

Section 3.1

(★) 3.1 (Checking Solutions) Check if the given value(s) is/are solutions to the following equations.

(★) 3.2 (Equation Diagrams) Write the equation that best represents the following diagrams.

(★) 3.3 (Scales) Look at the diagrams below. How many circles would you need to balance the third scale?

(★) 3.4 (Scales) Which is heavier? A red triangle or a blue square?

Section 3.2

(★) 3.5 (Finding Inverses) Find the additive and multiplicative inverses of the following numbers.

1. $\ -6$
2. $\ 4{\frac {2}{3}}$
3. $\ -0.33$
4. $\ 2+{\sqrt {5}}$

(★) 3.6 (Thinking of a Number) I think of a number, then I multiply the number by 6. My answer is 48. What was the original number I started with?

(★★) 3.7 (Candy) Brian has some candy. He gives away ${\frac {3}{4}}$ of it, and then eats 3 pieces. He has 6 pieces of candy remaining. How many pieces of candy did he start with?

(★★) 3.8 (Missing Digits) Doris was performing some calculations, but spilt ink on her work. Use inverse operations to find the missing numbers.

1.

${\begin{array}{r}4X3\\+\ 24X\\\hline 674\end{array}}$

2.

${\begin{array}{r}39X1\\+\ X3X\\\hline 4176\end{array}}$

3.

${\begin{array}{r}X45\\-\ 32\\\hline 91X\end{array}}$

Section 3.3

(★) 3.9 (Solving Equations) Solve the following equations for x.

(★★) 3.10 (Find A) If $2x+1=a$ and $3x-2=a$, find the value of $a$

(★★) 3.11 (Number of Solutions) Refer to the equation $3x-8=ax+7$. For what values of $a$ does the equation have:

1. a unique solution?
2. no solution?
3. infinitely many solutions?

Section 3.4

(★) 3.12 (Baseball Cards) Steven bought 4 packs of baseball cards, each with the same number of cards inside. If Steven had 32 cards overall, how many cards were in each pack?

(★) 3.13 (Oxygen) Approximately 21% of air is oxygen. Using this estimate, determine how many liters of oxygen are in a room containing 25,400 L of air.

(★) 3.14 (Triangle) A triangle has a 105 degree angle and a 32 degree angle. What is the measure of its third angle?

(★) 3.15 (Barrels of Water) 4 barrels of equal volume are filled with water. 3 of the barrels are completely filled and the fourth barrel is filled with only 8 gallons. If 53 gallons of water have been pumped into the barrels overall, what is the volume of each barrel?

(★) 3.16 (Employee Payment) Jason pays each of his employees equally. If Jason had to pay each employee $50 and he gives them each a 10% bonus, how many employees does Jason have if he pays $440 overall?

(★★) 3.17 (Two Cars) A car travels 20 km/h faster than a second car. The first car covers 180 km, the second car covers 135 km in the same time. What is the average speed of each car?

(★★) 3.18 (Equal Wages) Quinton and Thomas work at a company. Quinton's wages are $20 per hour, while Thomas has wages of $40 per hour. Quinton, however, being in sales, can expect a $3 raise (per hour) every month, while Thomas can only expect a raise of 50 cents. How long until they are paid the same amount per hour?

(★★) 3.19 (Sum of Interior Angles) The sum of the interior angles of a polygon can be found by subtracting 2 from the number of sides and multiplying the result by 180°.

1. Write an equation that represents the sum of the interior angles s of a polygon with n sides.
2. Use your equation to determine the total interior angle sum of an octagon.
3. A regular polygon is a polygon that has equal side lengths and equal interior angles. What is the value of each interior angle in a regular hexagon?
4. The sum of the interior angles for a particular polygon is 1440 degrees. How many sides does this polygon have?
5. For a given regular polygon, the measure of one interior angle is 156 degrees. How many sides does it have?