Math College Algebra

CHAPTER 1 Which family of numbers begins with the numbers 0, 1, 2, 3, …?

Whole numbers Whole numbers include the counting numbers 1, 2, 3, and so on plus the number 0.

Consider the numbers 1/2 and 9. Which of the following statements is completely true regarding the most specific classification of these two numbers?

1/2 is a rational number and 9 is a natural number. The most specific classification for the number 1/2 is a rational number as it is not an integer. The number 9 is an integer, but can be more specifically classified as a natural number.

Which of these statements is FALSE regarding the number 0?

0 is the smallest natural number. 0 is a real, rational number. It is also the smallest whole number. It is not a natural number, though.

Which of the numbers in the following set are rational numbers?

500, -15, √2, 1/4, 0.5, -2.50, π

500, -15, 1/4, 0.5, -2.50 Rational numbers include integers, natural numbers and whole numbers. √2 and π are not rational numbers. The rest are.

Which of the following is an example of an ordered pair?

(2,9) An ordered pair is defined as a set of coordinates shown as two numbers inside parentheses.

What are the coordinates of this point?

(-3, 4) Ordered pairs always follow the (x,y) format. The point is located -3 on the x-axis and 4 on the y-axis.

What is the special name for the coordinate (0,0)?

Origin The point at the very middle of the graph is called the origin, and its coordinates are (0, 0), because it's 0 units away from the center of the graph in both directions.

Which graph has a positive x-intercept and a negative y-intercept?

The line crosses the x-axis to the right of the origin, making it positive, and it crosses the y-axis beneath the origin, making it negative.

The point (1, -5) would be found where on the Cartesian plane?

In the lower-right quadrant On the Cartesian plane, Quadrant 4 has the x value positive and the y value negative.

If the equation y = 50x + 25 represents the amount of (y) money it costs to rent a car for (x) days, how long could you rent a car for if you had 775 dollars?

15 days Plugging in 775 for y and solving, you get this:

775 = 50x + 25

750 = 50x

x = 15

For which of these line equations is the WRONG slope given?

3x + y = 15; m = 3 Put each of the equations in slope y-intercept form, y = mx + b

For 3x + y = 15; m = - 3

y = -3x +15, so the slope is -3 and not 3.

Which of the following represents a linear equation that begins at 5 and increases by 8 at each step?

y = 8x + 5 Using the slope-intercept form, the number by itself is the y-intercept, or where it begins, and the number in front of the x is the slope, which tells you how much it increases with each step. So, for this problem, the number by itself needs to be 5, and the number in front of the x needs to be 8.

Which graph represents a linear equation?

The one with the straight line represents a linear equation.

Which pattern of numbers represents a linear equation?

3, 9, 15, 21, 27, 33 A linear graph increases by the same amount each time, so the pattern 3, 9, 15, 21, 27, 33 is linear as it adds 6 with each step. The pattern 2, 4, 8, 16, 32, 64 may look like the right answer, but it is exponential, 2n.

What is the equation of the graph below?

y = 2x + 8 By substituting either the x and y intercept point, (-4,0) or (0, 8) into the five given equations, we can determine which one is correct: -2(-4) + 0 = 8 and -2(0) + 8 = 8

Which linear equation is shown below?

y = 5x + 2 To find the linear equation for a given graph, pick two points on the line (x1, y1; x2, y2) and then calculate the slope between the points. For example, the slope between 0, 2 and -1, -3 is:

• m = (y2 - y1) / (x2 - x1)

• m = (-3 - 2) / (-1 - 0)

• m = -5 / -1

• m = 5

Therefore: y = 5x + b.

Solving for b (0, 2):

• 2 = 5 * 0 + b

• b = 2 - (5 * 0)

• b = 2

Solving for b (-1, -3) again:

• -3 = 5 * -1 + b

• b = -3-(5 * -1)

• b = 2

Therefore, the linear equation for the graph is y = 5x + 2.

Which slope intercept form equation is equivalent to the standard form equation -x + 2y = -8?

y = (1/2)x - 4 The slope-intercept form of a linear equation is y = mx + b.

• -x + 2y = -8 (add x to both sides of the equation)

• 2y = x - 8 (divide each number/variable by 2)

• y = (1/2)x - 4

What is the slope-intercept form equation of the graph below?

y=-1/2x+2 Using our linear equation model y = mx + b, We know that the y-intercept is 2 and that the slope is negative, going down one point for every two points it moves to the left. This leaves us with y = -(1/2)x + 2.

Which line has the greatest slope?

The greatest slope is the line that has the greatest vertical movement with the least horizontal.

If the graph below was written in the slope-intercept form, which of the following would be true?

m>0, b<0 Because the x-intercept is a positive number and the y-intercept is a negative number, we know that m is greater than zero and b is less than zero.

If the graph below has the equation Ax + By = C, which statement MUST be true?

C = 0 While there could be many things true about A or B, C must be zero in this example in order for the line to go through the origin.

If the graph below has the equation Ax + By = -4, which of these statements is true about A and B?

A>0, B>0 If we look at the graph both the x and y intercepts are negative. But, that's not what the question is asking. We are looking at A and B, the coefficients of x and y.

At the x intercept y = 0. If we substitute this into our equation (Ax + By = -4), we get Ax = -4. Solving for x: x = -4/A. We know that the x intercept is negative, so A has to be a positive number. (If we were to divide -4 by a negative number, we would get a positive number for the x intercept.) Therefore, A > 0.

Similarly, at the y-intercept, x is equal to zero. Substituting this into the equation Ax + By = -4 gives us By = -4. Solving for y: y = -4/B. We know from the graph that the y-intercept is negative. So, B has to be positive as well. Therefore B>0.

How would you express a line with a zero slope?

y = 0x + b Recall that the slope-intercept form is y = mx + b. To express a line with a zero slope, we'd replace 'm' with a zero.

Which one of the following equations could work for the following graph?

y = 5 The red line is parallel with the x-axis and above the y = 0 mark. Therefore, y = 5 is the only possible option.

Which of the following graphs represents x = -3?

The x-point is -3, so it must run vertical and be to the left of the y-axis.

For which one of the following lines would the graph fall exactly on top of the x-axis?

y = 0 Any line above or below y = 0 would be parallel to the x-axis.

The slope for a horizontal line would be which one of the following?

0 Horizontal lines do not experience any vertical changes, making their slope 0.

What is the slope of a line that is perpendicular to y = 3x + 1?

-1/3 The slope of the line y = 3x + 1 is 3. A line that is perpendicular to y = 3x + 1 will have a slope opposite and reciprocal to 3, that is -1/3.

What is the equation of a line with slope of 2 and a y-intercept of -6?

y = 2x - 6 According to the question, the slope (m) = 2 and the y-intercept (b) = -6. Replace the two values in the general equation y = mx + b, which gives y = 2x - 6.

What is the equation of a line that is parallel to y = -4x + 2 and goes through the point (-4, 9)?

y = -4x - 7 Parallels lines have same slope. If a line is parallel to y = -4x + 2, it will have a slope of -4. Since the line passes through the point (-4, 9), the equation will be y = -4x - 7.

Which linear equation has the solutions (2, 4) and (-2, 6)?

y = (-1/2)x + 5 The two points (2, 4) and (-2, 6) give a slope of -1/2.

Using one of the two points and the value of the slope (-1/2) gives the line y = -1/2x + 5.

You can test both the points by substituting them into the equation. You should get a true math statement.

Which of these equations represents a line that is perpendicular to the line in the graph below?

y = -5x - 2 The slope of the graph in the image is 1/5. Any line that is perpendicular must have a slope = -5 (opposite and reciprocal). We have not been given a point for the perpendicular line to go through. Line y = -5x - 2 has a slope of -5, making it the correct answer choice.

Based on this graph, what is the solution to the system of equations?

(3, 2) The point where the lines meet provide the solution.

Solve the system of equations below:

y = 3x - 5

y = -2x + 10

(3, 4) Since both equations equal (y) we want to replace one (y) with the other equation, giving us a linear equation with one variable, -2x + 10 = 3x - 5. We then add 2x to both sides, eliminating the -2x and giving us 10 = 5x - 5. We then add 5 to both sides, giving us 15 = 5x. Finally, we divide both sides by 5 to get x = 3. We then place our x value into one of the original equations to give us y = 4.

Which graph does NOT represent a system of equations?

A system of equations is having more than one equation in a single problem.

Which of the following is a solution to the system of equations shown in the graph below?

There is no solution. The graph shows two parallel lines, which means that there is no intersecting point, and no solution.

Consider the following two car rental companies. Cars R Us charges $100 up front, plus 10 cents per mile. Thrifty Car Rentals charges 25 cents per mile with no upfront fee. If TC represent the total cost for the car, and n represents the number of miles, which of the following systems of equations best represents this situation?

TC = 100 + 0.10n

TC = 0.25n When we set up the equations for the two companies, we will be solving for the total cost of the rental (TC). By using a system of equations, we will get one value for both equations, the point where both companies charge the same amount.

Which of the following represents a valid first step to solving this system of equations using the substitution method?

2x - y = 7

x = 2y - 1

2(2y - 1)- y = 7 Because the second equation gives us the value of x, the first step would be to replace x with its determined value, or 2(2y - 1)- y = 7.

Because the second equation gives us the value of x, the first step would be to replace x with its determined value, or 2(2y - 1)- y = 7.

(6,-3) We want to use the elimination method for this problem. Since the y value already has opposite values, we'll use y as our elimination variable. Multiply the first equation by 4 and the second by 3, then add the two together, eliminating the y variable and leaving 23x = 138. Divide both sides by 23, giving us x = 6. Replacing the x in either of the starting equations gives us our final variable of y = -3. (6,-3).

Which equation represents a valid first step to solving the following system of equations using the elimination method if our next step is to add the equations together to eliminate the x variable?

2x - 2y = 5 and -4x + 5y = 11

2(2x - 2y) = 2(5) In order to eliminate the -4x in the second equation, multiply the first equation by 2: 2(2x - 2y) = 2(5) → 4x - 4y = 10. Then add the equations to eliminate the x.

When trying to figure out how many $5 burritos and $3 tacos to buy, which equation represents the situation where you have $100 to spend?

5B + 3T = 100 Because we know the values of each item and we have the amount we want to spend, we need to identify how many burritos and tacos we want to buy. Therefore, we would use the 5B + 3T = 100 equation.

A car company has $44,000 to spend on robots to put paint on new cars. There are 2 types of robots, some that cost $3,000 and some that cost $2,000. If the company is sure that it wants to spend all of its money on 16 new robots, how many of each should it purchase?

12 expensive ones, 4 cheap ones We know that the base equation reads 2000C + 3000E = 44000 dollars and E + C = 16 robots. Working with the robot equation, we can form an equivalent equation: E = 16 - C. We can then use our value of E in the dollars equation: 2000C + 3000(16 - C)= 44000. Next we can use inverse operations to get the C by itself. Distribute the 3000 and combine like terms: 48000 - 1000C = 44000. Undo the 48000 and the 1000 using division, leaving us with C = 4. Replace the value in the robot equation: E + 4 = 16. Subtract 4 from both sides gives 12 expensive ones and 4 cheap ones.

Subtract the following matrices:

[ 1 -2 4 -6] - [ 1 -3 3 -9]

[ 0 1 1 3] When subtracting two matrices, subtract those numbers that share the same position in the individual matrices:

• 1 - 1 = 0

• -2 -(-3) = 1

• 4 - 3 = 1

• -6 - (-9) = 3

Which system of equations is represented by the following matrix? (Think of the vertical line as the equal sign.)

[4 -1 1 5 8 7 6 8 -2 / 10 -5 1]

4x - y + z = 10

5x + 8y + 7z = -5

6x + 8y - 2z = 1 The first column gives the coefficients for the x variable, the second column gives the coefficients for the y variable and the third column gives the coefficients for the z variable. The last column gives the other side of the equal sign. So, this one gives the set of equations that begins with 4x - y + z = 10.

Add the following matrices:

[2 1 -2 4] + [-4 12 2 14]

[-2 13 0 18] To add the numbers in two matrices, add the numbers found in the same position in each matrix:

• 2 + (-4) = -2

• 1 + 12 = 13

• -2 +2 = 0

• 4 + 14 = 18

Which of these symbols is commonly used to enclose a matrix?

• [ ] Matrices uses square brackets that extend to the size of the matrix on either side.

How many rows would a matrix that represents a system of 5 variables and 3 equations have?

3 This matrix would have 6 columns for the 5 variables plus the constant and 3 rows, one for each equation.

Which of the following would be a square matrix?

6x6 A square matrix has the same number of rows and columns and can be represented by N x N.

Find the determinant of the following matrix.

[-3 4 2 1]

-11 The Determinant is (-3 x 1)-(4 x 2) = -11

Find the determinant of the following matrix.

[1 5 -2 9 3 1]

The determinant cannot be found as it is not a square matrix. The determinant cannot be found as it is not a square matrix

What size is the following matrix?

[2 -4 0 5 1 3 4 -3]

4x2 The above matrix have 4 rows and 2 columns so it is a 4x2 matrix.

Which choice below could be a possible answer to the determinant of a matrix?

-9 Matrices are grids of numbers. The only answer that is just a number is -9.

Absolute values make positive numbers _____ and negative numbers _____.

positive, positive Absolute values make all numbers positive no matter their sign.

What is the correct expression for the absolute value of the following number: 8?

|8| The absolute value of a number is expressed by placing a vertical bar on either side of it

Evaluate |5 - 3| + 4|3 - 5|

10 Evaluating inside the absolute value first: |2| + 4 |-2|. Applying the absolute value: 2 + 4 (2) = 2 + 8 = 10.

Evaluate |4 + 7| - |4 - 7|

8 Going by the order of operations, evaluate inside the absolute value first: |11| - |-3|. Then apply the absolute value: 11 - 3 = 8.

Evaluate |r + 4| - 3|s| when r = 5 and s = -1.

6 |r + 4| - 3|s| becomes |5 + 4| - 3|-1| Performing the operations in the absolute and then following the standard order gives 9 - 3 = 6.

If a question requires us to know the value of |2x + y|, but all we know is that |x + y| = 5, what can we do to solve?

There is nothing we can do with the equation. As we have to do whatever operations are on the inside of the absolute value, there is nothing we can do with this equation.

If a question requires us to know the value of y - x, but all we know is that x + 2 = y, what can we do to solve?

Subtract x from both sides of the equation. If a question requires us to know the value of y - x, but all we know is that x + 2 = y, you can subtract x from both sides of the equation.

Evaluate |r + s| - (r + s) when -s = r + 8.

16 |r + s| - (r + s) when -s = r + 8

The absolute value symbols contain the expression r + s, and so do the parentheses. If we can change the equation -s = r + 8 so that one side equals r + s, then we will know the value of r + s.

First, subtract 8 on both sides so the equation becomes -s - 8 = r. Then, add s on both sides to get -8 = r + s. We now know that r + s is equal to -8.

Since r + s = -8, we can replace r + s with -8 in the equation. The equation then becomes |-8| - (-8).

Absolute value symbols make the end result of whatever is inside positive, so the equation becomes 8 - (-8). Subtracting -8 is the same as adding 8. So, the final equation is 8 + 8 = 16

Absolute value symbols around a mathematical operation should be treated like which one of the following?

Parentheses Absolute value signs around a mathematical operation should be treated like parentheses, which means you we should perform the operations inside the symbols first.

Solve the equation: |x + 8| = -1.

No solution Remember that absolute values that equal negative numbers have no solution because absolute values are always positive.

In this case the absolute value is negative, so there is no solution.

What number can we substitute for x that will make the absolute value of x = 12?

12, -12 Absolute value equations always result in two answers, so we would plug in 12, and -12

Solve the equation: |x| = 99.

x = 99 or x =-99 Remember that absolute value indicates how far a number is away from 0. So x = 99 or -99.

Solve the equation: |3x - 9| = 27.

x = 12, -6 The solution to this equation is x = 12, -6:

• |3(12) - 9| = 27

• |3(-6) - 9| = 27

Solve the equation: |2x - 5| = 5.

x = 5 or x = 0

2x - 5

= 5 is:

• Either 2x - 5 = 5 or 2x - 5 = -5

• Either x = 5 or x = 0

Solve the equation: -2|x| = 10.

No solution Remember Onions!

-2|x| = 10 is the same as |x| = -5.

Since |x| should be positive, the above equation has no solution.

Solve the equation: 3|x - 2| - 1 = 11.

x = 6 or -2 Remember onions!

3|x - 2| - 1 = 11 simplifies to |x - 2| = 4

So, x - 2 = 4 or x - 2 = -4.

x = 6 or x = -2.

Solve the equation: |2x + 5| = 11.

x = 3 or x = -8 Either 2x + 5 = 11 or 2x + 5 = -11.

Solving for x in both equations yields: x = 3 or x = -8.

What is the first step to solving the equation |2x + 1| - 5 = 7?

Add 5 to both sides. The last step performed to set the equation was to subtract 5.

So, the first step in solving the equation would be to add 5 to both sides.

What are the coordinates of the vertex of the graph y = |x - 9| - 2?

(9, -2) An absolute value transforms negative numbers back into positive ones, so x - 9 becomes 9, and the y value begins at -2, yielding (9,-2)

Which equation matches the graph below?

y = | x + 4 |

Which graph matches the equation below?

y = | x | - 3

Following the rules of absolute value, the vertex point would read (0,-3).

Which graph below could represent an absolute value?

All absolute value graphs more or less look like this, the letter V.

In the equation y = 3|x-4| + 7, what are the coordinates of the vertex?

(4, 7) The absolute value will flip the negative within the brackets, making (4,7) the answer.

A graph of y = (1/3)|x| would exhibit which of the following characteristics.

A wide dilation A transformation that makes the graph wider or thinner is called a dilation.

Which equation describes the following graph?

y = - 1/3 | x - 4 | + 2 Since the graph is mirrored, we know that the first number must be negative and that the slope is -1/3. The vertex is located at (4,2), so the absolute value must be x - 4 and have a y value of +2.

Which graph describes the following equation?

y = 2 | x + 1 | - 4

Since the first number is positive, we know that the lines will be moving up the y axis. Here, x + 1 shows us that the x value will be at -1, while the the -4 tells us that the point is at -4. This makes our vertex point (-1,-4)

Which of the following is a type of inequality sign?

All of the answers are correct. <, >, ≥ and ≤ are all inequality signs.

Which statement would be represented by an inequality?

To ride the roller coaster you must be at least four feet tall. The only statement that introduces a range of possible values is the roller coaster requirement of at least four feet tall. So anyone four feet tall or taller can ride.

Typically, an inequality can have how many solutions?

Multiple solutions Inequalities generally give you a possible range of answers, so you will have multiple solutions.

Which of the following is an example of an inequality?

60 + 2x < 120 Inequalities always have an inequality sign instead of an equal sign.

Inequalities can appear in which of the following types of equations?

All of the answers are correct. Inequalities can appear in quadratic, logarithmic, rational, linear, cubic and exponential equations.

Which inequality is graphed below?

y > 2x - 4 The y-intercept is -4 and the slope is 2.

The line is dotted, so the inequality is either 'greater than' or 'less than'. Use a test point that lies in the shaded area to figure out the inequality.

What makes solving inequalities different from solving an equation with an equal sign? For an inequality, you must _____.

flip the inequality sign when multiplying or dividing by a negative number. When solving inequalities you must flip the sign when multiplying or dividing by a negative number.

Solve and graph the following inequality:

2x + 5 > 11.

When determining the smallest number we can use to make the equation true, we find that 3 will equal 11. Since the inequality states that it is greater than, but not equal to, we must select the number line that has an open circle on the 3 with a right running arrow (red).

A mathematician goes shopping and sets his budget (in dollars) using the following set notation {x | x > 50 and x < 75} where x is the amount he will spend. Which of the following is within his budget?

$53 The number must be greater than 50, and less than 75. $53 is the only number that fits those parameters.

Which sentence best describes an 'OR' compound inequality?

A problem with two 1-variable inequalities where only one condition must be satisfied in order to be a solution A compound inequality where only one condition must be satisfied is called an 'or' compound inequality.

Everyone in your family is older than 3 but younger than 97. Express this fact using interval notation.

(3, 97) Interval notations are set up using parentheses. Because the range does not include 3 or 97, the set is bound by parentheses.

Which sentence best describes the 'AND' compound inequality?

A problem with two 1-variable inequalities where the conditions in both must be satisfied in order to be a solution A compound inequality where both conditions must be satisfied is called an 'and' compound inequality.

Solve the following inequality: x + 2 > 10.

x > 8 Here, we move all of the terms that do not contain an x to the right side of the inequality (10- 8 = 2).

This shaded portion of the graph corresponds to which inequality?

y ≥ x + 4 The solid line means the line is included, so we have an inequality that is either ≥ or ≤. Since it is above the line, it should be ≥.

To test this, try the point (0,0). This gives 0 ≥ 4, which is not true. That means that the point (0,0) is not included in the answer set, so the shaded region should be above the line. If you test the point (0,5), which is in the shaded region, you get 5 ≥ 4, which is true.

What part of this graph is the solution area for a system of inequalities?

The overlapping part When you have a system of inequalities, the part of the graph that is the solution area is the overlapping part where values are true for both inequalities.

Solve the following inequality: 2x + 4 < 20.

x < 8 To solve this problem, we divide each side of the inequality by the factors that do not contain the variable:

• Subtract 4 from both sides of the inequality: 2x < 16

• Divide both sides of the inequality by 2: x < 16/2

• x < 8

Which rule do you need to remember when solving a 1-variable absolute value inequality?

When undoing an absolute value, you must flip the inequality symbol on the negative solution. When solving a 1-variable absolute value inequality, while undoing an absolute value, you must flip the inequality symbol on the negative solution.

When the arrows of an inequality go in opposite directions, or the absolute value is greater than, what type of a compound inequality are we working with?

Or An 'or' compound inequality is when the arrows of an inequality go in opposite directions.

Solve the following inequality:

| x + 6 | <- 4

-10 ≤ x ≤ -2 To solve this problem:

• Rewrite the inequality without the absolute value: -4 ≤ x + 6 ≤ 4.

• Subtract 6 from the equation: - 4 - 6 ≤ x ≤ 4 - 6.

• Simplify: -10 ≤ x ≤ -2

Graph the following inequality: |x| > 2

Because x can be positive or negative, the graph has to be greater than 2 and less than -2, but not include either 2 or -2. The graph would have open circles at 2 and -2 with arrows moving away in opposite directions from both circles.

Which of the following inequalities does not have a solution?

|x| + 4 < 0 If we look at:

|x| + 4 < 0

and solve:

|x| < -4

It is impossible for us to have an absolute value smaller than -4; therefore, there are no solutions to this one.

What does a closed dot on a number line signify?

The number is included. A closed dot on a number line means that the number is included.