Download Compound Inequality Answers Compound Inequality Explanations

Algebraic Concepts - Part 2
Compound Inequality Answers
1.
C
2.
B
3.
D
4.
A
5.
A
6.
A
7.
B
8.
A
9.
A
10.
B
11.
A
12.
D
13.
C
14.
D
15.
B
Compound Inequality Explanations
Algebraic Concepts - Part 2
1.
3x - 4 < -10
OR
3x - 5 > 1
3x < -6
OR
3x > 6
x < -2
OR
x > 2
Since the inequalities cannot be combined, the graph of the solution goes in opposite directions as
shown below.
-10
2.
-8
-6
-4
-2
0
2
4
Start by solving each of the inequalities individually.
-4x + 5 < 37
AND
-4x < 32
6
8
10
12
14
2x - 4 < 2
2x < 6
AND
x > -8
x < 3
AND
This gives the correct answer. However, this can be rewritten in a much better form.
x > -8
x < 3
AND
x < 3
-8 < x
AND
Now combine the two inequalities to get the final answer.
-8 < x < 3
3.
To solve the inequalities, isolate the
in the middle. Remember that whatever is done to the expression
in the middle must also be done to both numbers on the outside of the inequalities.
4.
2x + 1 < -3
OR
4x - 6 > 10
2x < -4
OR
4x > 16
x < -2
OR
x > 4
Since the inequalities cannot be combined, the graph of the solution goes in opposite directions as shown
below.
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
Algebraic Concepts - Part 2
5.
The number line represents numbers that are less than -4 and greater than 4.
Also, since the circles on -4 and 4 are filled in, -4 and 4 are included with the other numbers represented
by the number line.
So, the number line represents the following inequalities.
x < -4 and x > 4
This is equivalent to the inequality |x| > 4.
6.
Solve the inequality to determine the correct graph.
|2x - 6| < 2
-2 < 2x - 6 < 2
4 < 2x < 8
2<x<4
The correct graph is shown below.
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7.
To solve the inequalities, isolate the
in the middle. Remember that whatever is done to the expression
in the middle must also be done to both numbers on the outside of the inequalities.
8.
Start by solving each of the inequalities individually.
-6x + 6 < 42
AND
-6x < 36
AND
4x - 1 < 7
4x < 8
x > -6
x < 2
AND
This gives the correct answer. However, this can be rewritten in a much better form.
x > -6
x < 2
AND
-6 < x
AND
x < 2
Algebraic Concepts - Part 2
Now combine the two inequalities to get the final answer.
-6 < x < 2
9.
The number line represents numbers that are greater than -4 and less than 4.
Also, since the circles on -4 and 4 are open, -4 and 4 are not included with the other numbers
represented by the number line.
So, the number line represents the following inequalities.
x > -4 and x < 4
This is equivalent to the inequality |x| < 4.
10.
When given a double inequality to solve, focus on solving for the x in the middle. Remember that
whatever is done to the expression in the middle must also be done to both the numbers on the outside of
the inequalities.
-14 < 3x + 1 < 7
-15 <
3x
< 6
-5 <
x
< 2
11.
Since x > -5 and x < 2, the graph will have an open circle at x = -5 and a closed circle at x = 2.
Since the compound inequality is an "AND" inequality, or a conjunction, the graph of the inequality will be
shaded between x = -5 and x = 2.
Therefore, the graph of the given inequality is shown below.
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
12.
The situation can be set up as a compound inequality as shown below.
$80 < 3x - 4 < $167
The compound inequality above can be separated into two inequalities.
After being separated, solve the two inequalities for x.
3x - 4 < 167
80 < 3x - 4
and
3x < 171
84 < 3x
and
x < 57
28 < x
and
These two answers can be combined as shown below.
28 < x < 57
This compound equality solved for x shows that the least and most amount she can spend per pair of
shoes is $28 and $57 respectively.
13.
Algebraic Concepts - Part 2
-4 < 2x + 2 < 22
-6 <
2x
< 20
-3 <
x
< 10
The number line that represents this solution is shown below.
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
14.
The situation can be set up as a compound inequality as shown below.
134 < 8x + 6 < 182
The compound inequality above can be separated into two inequalities.
After being separated, solve the two inequalities for x.
8x + 6 < 182
134 < 8x + 6
and
8x < 176
128 < 8x
and
x < 22
16 < x
and
These two answers can be combined as shown below.
16 < x < 22
This compound equality solved for x shows that the range for the number of round tables she will need for
her reception is 16 to 22.
15.
Solve each of the inequalities individually to get the correct answer.
4x - 4 > -20 OR -5x < -35
4x > -16
OR
x > 7
x > -4
Since x > -4 includes all the values of x such that x > 7, the correct answer is x > -4.
Algebraic Concepts - Part 2
Graph Linear Equations & Inequalities
Answers
1. B
2. D
3. C
4. A
5. A
6. B
7. D
8. C
9. A
10. C
11. C
12. D
13. A
14. D
15. D
Graph Linear Equations & Inequalities
Explanations
1. All the answer choices are written in slope-intercept form, y = mx + b, where m is the slope and b is
the y-intercept.
Algebraic Concepts - Part 2
The graph passes through the point (0,-2). Thus, the y-intercept is -2.
From the y-intercept (0,-2), move up 3 spaces and 1 space to the right to the point (1,1). This gives
the following slope.
So, the equation of the line is given below.
2. All the answer choices are written in slope-intercept form, y = mx + b, where m is the slope and b is
the y-intercept.
The graph passes through the point (0,1). Thus, the y-intercept is 1.
From the y-intercept (0,1), move up 1 space and 5 spaces to the right to the point (5,2). This gives the
following slope.
So, the equation of the line is given below.
3.
4.
Algebraic Concepts - Part 2
5.
6.
7.
Algebraic Concepts - Part 2
8.
9.
10. All the answer choices are written in slope-intercept form, y = mx + b, where m is the slope and b is
the y-intercept.
The graph passes through the point (0,-3).
Thus, the y-intercept is -3.
From the point (0,-3), move up 3 spaces and 2 spaces to the right to end up at the point (2,0). This
gives the following slope.
So, the equation of the line is given below.
11. All the answer choices are written in slope-intercept form, y = mx + b, where m is the slope and b is
the y-intercept.
The graph passes through the point (0,3). Thus, the y-intercept is 3.
From the y-intercept (0,3), move up 2 spaces and 1 space to the right to the point (1,5). This gives the
following slope.
So, the equation of the line is given below.
Algebraic Concepts - Part 2
12.
13. All the answer choices are written in slope-intercept form, y = mx + b, where m is the slope and b is
the y-intercept.
The graph passes through the point (0,-2).
Thus, the y-intercept is -2.
From the point (0,-2), move up 2 spaces and 3 spaces to the right to end up at the point (3,0). This
gives the following slope.
So, the equation of the line is given below.
14.
15.
Algebraic Concepts - Part 2
Solve Linear Equations Answers
1. B
2. D
3. D
4. D
5. B
6. C
7. C
8. D
9. C
10. B
11. C
12. B
13. A
14. C
15. B
Solve Linear Equations Explanations
1.
Algebraic Concepts - Part 2
2.
3.
Let C(x) represent the daily production cost and x represent the number of pairs of skis
manufactured. The daily production cost is equal to the cost per pair of skis plus the fixed costs.
So,
daily production cost = (cost per pair)(# of pairs) + fixed costs
Now, find the cost per pair in order to determine the equation for the daily production cost.
C(x) =
9,900 =
9,000 =
450 =
(cost per pair)x + 900
(cost per pair)(20) + 900
(cost per pair)(20)
cost per pair
The equation for the daily production is given below.
C(x) = 450x + 900
4.
5.
This situation can be represented by a linear equation where the total amount paid on the loan
is the dependent variable and number of monthly payments made is the independent variable.
Algebraic Concepts - Part 2
The amount already paid toward the loan is the y-intercept, and the amount paid each month is
the rate of change, or slope.
Use the given information to develop an equation and solve for the number of months.
y
$12,700
$1,700
17
=
=
=
=
mx + b
$100x + $11,000
$100x
x
It will take Kirk 17 more months to finish paying back the loan.
6.
Let y represent the total amount of the purchase and x represent the number of gallons of gas.
y(x) = (cost per gallon)x + carwash
y(x) = 2.46x + 7
Since she purchased 10 gallons of gas, her total purchase costs:
y(10) = 2.46(10) + 7 = $31.60
7.
8.
Let x represent the cost of the Mustang and y represent the total cost after tax. The total cost
of the Mustang is equal to the cost of the Mustang itself plus the tax.
total cost = Mustang price + tax
y = x + 0.08x
y = (1 + 0.08)x
y = 1.08x
9.
Find C(290).
C(290) = 0.6(290) + 40 = 174 + 40 = $214
Algebraic Concepts - Part 2
10.
This situation can be represented by a linear equation where the monetary goal for the month
is the dependent variable and the number of sales made is the independent variable.
The base salary is the y-intercept, and the amount earned per sale is the rate of change, or
slope.
Use the given information to develop an equation and solve for the number of sales.
y
$4,760
$3,360
84
= mx + b
= $40x + $1,400
= $40x
=x
Sheila needs to make 84 sales in order to reach her goal.
11.
12.
The problem states that the plane leaves the ground at 3,958 feet above sea level, and then
climbs 456 feet per minute.
So, set up the function with the plane's cruising altitude equal to the plane's beginning altitude
plus the climbing rate in t minutes from take off.
Cruising Altitude = Initial altitude + (rate of climb per minute)(minutes)
13,990 = 3,958 + 456t
13.
This situation can be represented by a linear equation where the total number of typed words
is the dependent variable and the number of minutes typing is the independent variable.
The number of words already typed is the y-intercept, and the number of words typed per
Algebraic Concepts - Part 2
minute is the rate of change, or slope.
Use the given information to develop an equation and solve for the number of minutes typing.
y
1,770
880
16
= mx + b
= 55x + 890
= 55x
=x
She has 16 minutes of typing left.
14.
The blender is completely depreciated when S(t) = 0. So, set S(t) = 0 and solve for t.
0 = -5,250t + 42,000
5,250t = 42,000
t=8
15.
Algebraic Concepts - Part 2
Systems of Equations Answers
1.
2.
D
D
3.
D
4.
5.
D
B
6.
C
7.
D
8.
D
9.
A
10.
B
11.
C
12.
C
13.
A
14.
C
15.
B
Systems of Equations Explanations
1.
The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (5, -5).
Algebraic Concepts - Part 2
So, x = 5 and y = -5 is the solution.
2.
3.
4.
First, set up equations to represent the two water packages.
Let x represent the volume of water used in thousands of gallons. Let y represent the yearly cost
of the plan.
Pay per thousand gallons:
y = ($0.75/thousand gallons)(x thousand gallons)
y = $0.75x
Pay fixed amount:
y = $825,000
Next, set the equations equal to each other, and solve for x.
$0.75x = $825,000.00
x = 1,100,000
Therefore, an industrial user needs to use over 1,100,000 thousand gallons of water a year for
the fixed amount package to be the less expensive option.
5.
First, set up equations that represent the cooking time and the weight of the turkey and the
ham.
Let t represent the weight of the turkey, in pounds, and h represent the weight of the ham, in
pounds.
Algebraic Concepts - Part 2
Cooking time:
(16 min/lb)(t lbs) + (23 min/lb)(h lbs) = 440 min
16t + 23h = 440
Weight:
t lbs = 2(h lbs)
t = 2h
Next, substitute 2h in for t in the first equation, and solve for h.
16(2h) + 23h
32h + 23h
55h
h
=
=
=
=
440
440
440
8
Then, substitute 8 in for h in the second equation, and solve for t.
t = 2(8)
= 16
Therefore, the turkey weighs 16 pounds, and the ham weighs 8 pounds.
6.
7.
8.
9.
Remember that x represents the number of hours Sydney spends working at the gift shop and
y represents the number of hours Sydney spends babysitting.
First, write an equation for the money Sydney earns and spends on her father's birthday
present.
Algebraic Concepts - Part 2
$7x + $4y - $35 = 0
Next, write an equation for the amount of time Sydney spends working, and rearrange it so that
one side is equal to zero.
x + y = 14
x + y - 14 = 0
Put these together to make a system of equations.
$7x + $4y - $35 = 0
x + y - 14 = 0
10
.
11
.
12
.
The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (1, 1).
So, x = 1 and y = 1 is the solution.
13
.
Remember that x represents the number of nights spent at the resort and y represents the total
cost of the trip.
Total Cost = Total Cost of Resort + Cost of Airfare
Total Cost = (Cost of Resort per Night × Number of Nights) + Cost of Airfare
Substitute in the information given in the problem for each vacation package.
In the first vacation package, round-trip airfare costs $870, and it costs $151 per night to stay at
the resort.
y = $151x + $870
In the second vacation package, round-trip airfare costs $252, and it costs $654 per night to
Algebraic Concepts - Part 2
stay at the resort.
y = $654x + $252
The following system of equations could be used to find how many nights Paul needs to stay at
either resort so that both vacation packages have the same cost.
y = $151x + $870
y = $654x + $252
14
.
First, set up equations to represent the amount of money Paulo spent on sale items and the
number of items he bought.
Let s represent the number of sale items he bought, and let r represent the number of regular
items he bought.
Money spent:
$52.02 + $40.87 = $56.07 - ($0.15/item)(s items) + $42.87 - ($0.10/item)(r items)
$92.89 = $98.94 - $0.15s - $0.10r
$0.15s + $0.10r = $6.05
Number of items:
s=r+7
Next, substitute r + 7 in for s in the first equation, and solve for r.
$0.15s + $0.10r
$0.15(r + 7) + $0.10r
$0.15r + $1.05 + $0.10r
$0.25r + $1.05
$0.25r
r
=
=
=
=
=
=
$6.05
$6.05
$6.05
$6.05
$5.00
20
Then, substitute 20 in for r in the second equation, and solve for s.
s = r+7
= 20 + 7
= 27
Since Paulo bought 27 sale items and 20 regularly priced items, he bought
27 + 20 = 47 items in all.
15
.
The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (4, 2).
So, x = 4 and y = 2 is the solution.