Algebra 1A Study Guide (with Problems and Solutions)

This study guide contains two sections--Section 1: Problems to Solve and Section 2: Solutions. First, solve the problems in Section 1 and then use Section 2 to check your work.

Section 1: Problems to Solve

Question 1:

David is rowing a boat upstream. The river is flowing at a speed of 2 miles per hour. David starts rowing at a speed of 6 miles per hour, but as he gets tired his speed decreases (at a rate of 1 mile per hour, every hour). What is the equation that represents the speed of the boat for x hours spent rowing?

Question 2:

What is the equation of the given line in the point-slope form?

Question 3:

What is the equation of a line passing through (-3, 4) and having a slope of -3?

Question 4

Solve and graph 5x - 22 < 8.

Question 5

4 times the square of a nonzero number is equal to 48 times the number. What is the number?

___________________________________

Question 6

Your teacher is giving you a test worth 100 points containing 40 questions.

There are two-point and four-point questions on the test. How many of each

type of question are on the test?

Question 7

The twins are having a graduation party. They bought hats for the boys and anklets for the girls for party bags. The hats cost twice as much as the anklets. They bought anklets for 30 girls and hats for 20 boys. The total spent was $227.50 How much was each hat? How much was each anklet?

Question 8

Solve this system of a equations by any method (Substitution/Elimination/Graphing- You pick)

30a+ 20h = 227.50h = 2a

Question 9

Given equations A and B as and 7 3

X+Y =15 and Y X = 10

3 7

, respectively, what would you multiply B by to eliminate the variable x?

Question 10

If the range of the function f(x) = 4x 3 is {11.4, 15, 17, 29}, what is its domain

Question 11

The money collected from selling bacon at a butcher store is given by the function f(x) = 3.55x 4, where f(x) is the sales revenue in dollars and x is the number of customers visiting the store each day. If {17, 21, 24, 34} customers visited over four days, what is the income from bacon sales each day?

Question 12

After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 minutes per day. How many weeks will it be before you are up to jogging 60 minutes per day?

Question 13

Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...

Question 14

Find the 20th term of the arithmetic sequence:-6, -10, -14, -18

Question 15

What is the solution of this linear system?

2x - 4y = -44-6x + 4y = 68

Question 16

Solve the following system of equations by using linear combinations.

8x - 8y = 1616x - 7y = 50

Question 17

Which ordered pair is a solution of

5x - 2y 6 ?

A. (0, -3)

B. (5, 5)

C. (1, -2)

D. (3, 3)

Question 18

If the nth term of a sequence is 2n + 4, what is the sum of the first n terms?

Question 19

If the first term of an arithmetic sequence is 25 and the 6th term is 100, what is the sum of the first 6 terms of the sequence?

Question 20

What is the sum of the first n terms of this sequence?

3, 4, 5, 6, 7...

Question 21

Which is the correct input-output table for the function f(x) = 7 4.5x?

Input (x)

1

3

4

6

10

Output f(x)

3.5

-3.5

-10

-18

-40

Input (x)

1

3

4

6

10

Output f(x)

2.5

-6.5

-11

-20

-38

Input (x)

1

3

4

6

10

Output f(x)

2

-4.5

-13

-21

-45

Question 22

A computer program starts with a node that branches out into 3 nodes, each of which repeats the branching process to form this tree structure:

Question 23

What is the 11th term of the geometric sequence -3,, , ,..?

Question 24

15x + 3y > 24 x + 3y -18

Question 25

Question 26

Given the function f(x) = x2, sketch the graph represents f(x + 4)?

Section 2: Solutions

Question 1:

David is rowing a boat upstream. The river is flowing at a speed of 2 miles per hour. David starts rowing at a speed of 6 miles per hour, but as he gets tired his speed decreases (at a rate of 1 mile per hour, every hour). Which equation represents the speed of the boat for x hours spent rowing?

(a) y = x + 4

(b) y = x 4

(c) y = 4 x [Answer]

Question 2:

What is the equation of the given line in the point-slope form?

Question 3:

What is the equation of a line passing through (-3, 4) and having a slope of -3?

Point-Slope Form (Memorize: POINT AND A SLOPE)

Given a slope m and a point P (x1, y1) on a line, the point-slope form of the equation of a line is given by:

y - y1 = m(x - x1)

y - y1 = m(x - x1)

Recall the formula for the point-slope form of a line.

y1 = 4x1 = -3m = 3

Label the known values for this problem.

y - 4 = 3(x - (-3))

Substitute the values into the formula. Be careful when one of the values in the point is negative. x minus a negative number is really going to be x plus that number.

y - 4 = 3(x + 3)

This equation is now in point-slope form.

y + -4 = 3x + 9

Put the equation in slope-intercept form.

y + -4 + 4 = 3x + 9 + 4

y = 3x + 13

The equation of a line with a point at (-3, 4) and a slope of 3 is given by y = 3x + 13.

Module Title

Linear Inequalities in 1 Variable, Part 1 (Alg1.1)

Question 4

Solve and graph 5x - 22 < 8.

5x < 30 x < 6 is the algebraic solution.

Graph: -6 0 6

Module Title

Using Linear Equations to Solve Problems (Alg1.1)

Question 5

4 times the square of a nonzero number is equal to 48 times the number.

What is the number?

(a) 12 [Answer]

(b) 24

(c) 96

(d) 4

Module Title

Solving Linear Systems Using Linear Combinations

Question 6

Your teacher is giving you a test worth 100 points containing 40 questions.

There are two-point and four-point questions on the test. How many of each

type of question are on the test?

Here you know there are two kinds of questions: The x questions and the y questions. It is a 40 question test, so the x questions and y questions put together makes 40 questions. x+ y= 40

The x questions count for 2 points each and the y questions count for 4 points each to make a total of 100 points. 2x + 4y =100

x+ y= 40

2x + 4y =100

You pick how to solve from here. One way would be solve for x by subtracting y from both sides of the first equation.

y = 40 x. Substitute into second equation and get:

2 (40 x) + 4y = 100

80 -2y +4y = 100

80 + 2y = 100

2y = 100

y = 10 There are 10 4 point questions.

Now figure out how many two point questions there are by plugging 10 for y in the first equation.

x + 10 = 40

x = 30 There are 30 2 point questions

Question 7

The twins are having a graduation party. They bought hats for the boys and anklets for the girls for party bags. The hats cost twice as much as the anklets. They bought anklets for 30 girls and hats for 20 boys. The total spent was $227.50 How much was each hat? How much was each anklet?

The hats cost twice as much as anklets means h=2a

If you buy 30 anklets and 20 hats it equals $227.50. 30a +20h= 227.50

Your system of equations is:

h=2a

30a +20h= 227.50 It is easier to use substitution because h is already alone.

30a + 20 (2a) = 227.50

30a + 40 a = 227.50

70 a = 227.50

a = $ 3.25 From the first equation, hats are double the amount of anklets.

h= $ 6.50

Question 8

Solve this system of a equations by any method (Substitution/Elimination/Graphing- You pick)If done algebraically with substitution:30a+ 20h = 227.50h = 2a

30a+20 (2a)= 227.5030a + 40 a = 227.50

70a = $227.50

a= $3.25 for anklets

hats are twice as much so h= 2 ($3.25)

So hats =$7.00

Question 9

Given equations A and B as and 7 3

X+Y =15 and Y X = 10

3 7

, respectively, what would you multiply B by to eliminate the variable x?

Answer: 7

A + B

3

Now you try:

Given equations A and B as and , respectively, which expression will eliminate the variable y?

(a) A + B

(b) A B

(c) A B

(d) A B [Answer]

(e) A + B

Question 10

If the range of the function f(x) = 4x 3 is {11.4, 15, 17, 29}, what is its domain?

(a) {1.5, 2.8, 5.5, 7}

(b) {2.3, 4, 8, 11}

(c) {7, 11, 13.2, 14}

(d) {3.4, 5, 8, 9.3}

(e) {3.6, 4.5, 5, 8} [Answer]

Question 11

The money collected from selling bacon at a butcher store is given by the function f(x) = 3.55x 4, where f(x) is the sales revenue in dollars and x is the number of customers visiting the store each day. If {17, 21, 24, 34} customers visited over four days, what is the income from bacon sales each day?

(a) {50.55, 63.45, 80.34, 99.8}

(b) {43.45, 58.75, 73.4, 93.5}

(c) {56.35, 70.55, 81.2, 116.7} [Answer]

(d) {45.74, 65.7, 83.8, 105.7}

(e) {63.25, 68.35, 79.7, 97.6}

Question 12

After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 minutes per day. How many weeks will it be before you are up to jogging 60 minutes per day?

Adding 6 minutes to the weekly jogging time for each week creates the sequence: 12, 18, 24, ...

This sequence is arithmetic.

Question 13

Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...

This is an arithmetic sequence with a common difference of 2.To find the sum, we need to know the last term (the 20th term).

Now we are ready to find the sum:

Question 14

Find the 20th term of the arithmetic sequence:-6, -10, -14, -18, . . .

an = a1 + (n - 1)d

Recall the formula for the nth term of an arithmetic sequence.

d = -10 - -6

Find the common difference by subtracting the first term from the second term in the sequence.

d = -4

The common difference for the sequence is -4.

a1 = -6n = 20d = -4

List the values for each variable.

a20 = -6 + (20 - 1)(-4)

Substitute the values into the formula.

a20 = -6 + (19)(-4)a20 = -6 + (-76)

Simplify.

a20 = -82

The value of the 20th term in the arithmetic sequence is -82

Module Title

Solving Linear Systems of Equations: Addition (Alg2.1)

Question 15

What is the solution of this linear system?

2x - 4y = -44-6x + 4y = 68

2x - 4y = -44-6x + 4y = 68

Align the equations. Make sure the like-terms and symbols line up.

-2x - 4y = -44+ -2x + 4y =- 68

Set up the equations in the form of an addition problem.

.2x - 4y = -44+ -2x + 4y =- 68

-4x + 0y =- 24

Add the equations together. The y terms cancel each other out because they are additive inverses of each other.

-4x = 24x = -6

Solve for x.

2x - 4y = -442(-6) - 4y = -44-12 - 4y = -44-4y = -32y = 8

Substitute x = -6 back into either of the original equations. The solution is x = -6, y = 8.

The solution is x = -6, y = 8.

Question 16

Solve the following system of equations by using linear combinations.

8x - 8y = 1616x - 7y = 50

Multiply top equation by 2

-16x + 16 y = -32

16x- 7y = 50

9y = 18

y =2

8x-8(2)= 16

8x-16 =16

8x= 32

x=4 Now check by plugging them both in to one of the original equations.

Question 17

Which ordered pair is a solution of

5x - 2y 6?

E. (0, -3) Correct

F. (5, 5)

C. (1, -2)

D. (3, 3)

Question 18

If the nth term of a sequence is 2n + 4, what is the sum of the first n terms?

(a) n(n + 1) + 4n [Answer]

(b)

(c) n(2n + 4)

(d)

(e)

Question 19

If the first term of an arithmetic sequence is 25 and the 6th term is 100, what is the sum of the first 6 terms of the sequence?

Sn =

n

[a1 + an]

2

Recall the formula for Sn.

a1 = 25a6 = 100n = 6

Write down the known information.

S6 =

6

(25 + 100)

2

Substitute the values into the equation for Sn.

S6 = 3(125)

Simplify.

S6 = 375

The sum of the first 6 terms of the arithmetic sequence is 375.

Question 20

What is the sum of the first n terms of this sequence?

3, 4, 5, 6, 7, ...

(a)

(b)

(c) [Answer]

(d) n(n + 1)

(e)

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Question 21

Which is the correct input-output table for the function f(x) = 7 4.5x?

(a)

Input (x)

1

3

4

6

10

Output f(x)

3.5

-3.5

-10

-18

-40

(b)

Input (x)

1

3

4

6

10

Output f(x)

2.5

-6.5

-11

-20

-38

[Answer]

(c)

Input (x)

1

3

4

6

10

Output f(x)

2

-4.5

-13

-21

-45

Question 22

A computer program starts with a node that branches out into 3 nodes, each of which repeats the branching process to form this tree structure:

If the program draws the tree to depth 5, how many nodes will the tree have?

(a) 36

(b) 40

(c) 80

(d) 82

(e) 121 [Answer]

Question 23

What is the 11th term of the geometric sequence -3, , , , ... ?

(a) [Answer]

(b)

(c)

(d)

(e)

Question 24

15x + 3y > 24 x + 3y -18

A system of inequalities is a group of inequalities that contain the same variables. For example, the following is a system of inequalities.

15x + 3y > 24 x + 3y -18

The solution to a system of inequalities are the values of x and y that made both inequalities true.

Earlier, you used graphing to solve systems of linear equations. The graph of the first equation in the system was a line, where all the points on that line made the first equation true. The graph of the second equation in the system was a line, where all the points on that line made the second equation true. The points that made both equations true, the solution, were where the two lines intersected.

It makes sense that the solution to a system of inequalities will also be where the solutions of the first and second inequality intersect. However, even though a system of inequalities may look similar to a system of equations, they are different. Since you are dealing with inequalities instead of lines, the graphs will not simply intersect at one point; instead, there will be two shaded areas on the coordinate plane. What will the intersection of two inequalities look like on a graph?

To find out, graph the two inequalities in the system on the same coordinate plane to see how they intersect.

15x + 3y > 24 x + 3y -18

To graph the first inequality in the system, begin by putting the first inequality into slope-intercept form.

15x + 3y = 243y = -15x + 24y = -5x + 8

The corresponding linear equation, y = -5x + 8, is the line part of the graph. This will be a dashed line because the initial inequality contains a greater than sign.

Use the point (0, 0) to figure out which side of the graph should be shaded.

15x + 3y > 2415(0) + 3(0) > 240 > 24

0 is not greater than 24, so the point (0, 0) is not a part of the solution. This means that all the points on the other side of the line are a solution to the first inequality.

This is the solution to the first inequality in the system of inequalities. Every point in the shaded region will make the first inequality true. However, the solutions to a system of inequalities are all the points which make both inequalities true. Graph the second inequality on the same coordinate plane to see where the two solutions intersect.

The second inequality must first be put into slope-intercept form.

x + 3y -18x + 3y = -183y = -x - 18

y = -

1

x - 6

3

The corresponding linear equation, y = -

1

x - 6, is the line part of the graph.

3

This will be a solid line because the initial inequality contains a greater than or equal to sign.

Use the point (0, 0) again to see which side of the line should be shaded.

x + 3y -18(0) + 3(0) -180 -18

Since 0 is greater than -18, all the points on the side of the line that has the point (0, 0) are solutions to this second inequality.

This is a graph of the system of inequalities:

15x + 3y > 24 x + 3y -18

The red shaded area contains the solutions that satisfy the first inequality in the system. The blue line and the blue shading are the solutions that satisfy the second inequality. Notice that the red shading and blue shading overlap in the purple area on the right side of the coordinate plane.

All the points in this overlapping area satisfy both the first and the second inequality. This is the graph of the solution to the system of inequalities; all the values of x and y in this area make both inequalities true. Because the blue line is solid, the points on the edge of the shaded area that are on the blue line are also included in the solution. The red line is dashed, so the points on the dashed line are not included in the solution.

Try a couple points in the shaded region to see if they make both the inequalities true. Also try a couple points that are not in the shaded region to see if they make the inequalities false.

Tips on Graphing Systems of Inequalities by Hand

1. Use a pencil! The solution to a system of inequalities should only contain the shading that is part of the solution. If you use a pencil, you can erase the extra shading that is not part of the solution without re-graphing the entire system.

2. The entire shaded area does not have to be filled in with solid shading. Most of the shading will be erased later on.

3. Shade lightly. This will make it easier to erase the shading that is not part of the solution.

4. Shade each inequality in a different way. This will make it clear where the shaded areas overlap. One good method is to use horizontal lines to shade one inequality and vertical lines to shade the other. Then the overlapping area can be seen.

5. Use the arrow method. If you don't have a pencil or a way to erase extra shading, use small arrows when graphing the inequalities to indicate which side of the line to shade. Then only shade in the area where all of the inequalities overlap in the final step.

Question 25

ii.

Using order of operations tells us that we should do what is inside the parentheses first and then deal with the exponent. To simplify within the parentheses involves working with several rules including the rule for negative exponents.

This step shows that the negative exponents were moved and exponents became positive.

This step shows combining exponents for terms that have the same base. Two different rules were used in this step: both the multiplication rule and the division rule.

This step is the final simplification of what is inside the parentheses. Now we have to raise each term in the parentheses to the power of 2.

It is not absolutely necessary to use this many parentheses, but it is useful in keeping track of each term that needs to be raised to the power of 2.

The final step is to simplify each term that has been raised to the 2nd power. It requires using the power rule for exponents.

This is a very common simplification problem. It makes extensive use of all the rules of exponents and requires several steps to get to the final answer.

Translations and Transformations (Alg2.2)

Question 26

Given the function f(x) = x2, which graph represents f(x + 4)?

(c) [Answer]

26