Algebra - PSAT Math
Card 0 of 5075
(√(8) / -x ) < 2. Which of the following values could be x?
(√(8) / -x ) < 2. Which of the following values could be x?
The equation simplifies to x > -1.41. -1 is the answer.
The equation simplifies to x > -1.41. -1 is the answer.
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Half of one hundred divided by five and multiplied by one-tenth is __________.
Half of one hundred divided by five and multiplied by one-tenth is __________.
Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.
Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.
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Let x&y be defined as (x – y)xy . What is the value of –1_&_2?
Let x&y be defined as (x – y)xy . What is the value of –1_&_2?
We are told that x&y = (x – y)xy .
–1&2 = (–1 – 2)(–1)(2) = (–3)–2
To simplify this, we can make use of the property of exponents which states that a– b = 1/(ab ).
(–3)–2 = 1/(–3)2 = 1/9
The answer is 1/9.
We are told that x&y = (x – y)xy .
–1&2 = (–1 – 2)(–1)(2) = (–3)–2
To simplify this, we can make use of the property of exponents which states that a– b = 1/(ab ).
(–3)–2 = 1/(–3)2 = 1/9
The answer is 1/9.
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The rate of a gym membership costs p dollars the first month and m dollars per month every month thereafter. Which of the following represents the total cost of the gym membership for n months, if n is a positive integer?
The rate of a gym membership costs p dollars the first month and m dollars per month every month thereafter. Which of the following represents the total cost of the gym membership for n months, if n is a positive integer?
The one-time first-month cost is p, and the monthly cost is m, which gets multipled by every month but the first (of which there are n -1). The total cost is the first-month cost of p, plus the monthly cost for (i.e. times) n -1 months, which makes the total cost equal to p + m (n -1).
The one-time first-month cost is p, and the monthly cost is m, which gets multipled by every month but the first (of which there are n -1). The total cost is the first-month cost of p, plus the monthly cost for (i.e. times) n -1 months, which makes the total cost equal to p + m (n -1).
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- If f(x) = (x + 4)/(x – 4) for all integers except x = 4, which of the following has the lowest value?
- If f(x) = (x + 4)/(x – 4) for all integers except x = 4, which of the following has the lowest value?
Plug each value for x into the above equation and solve for f(x). f(1) provides the lowest value –5/3
Plug each value for x into the above equation and solve for f(x). f(1) provides the lowest value –5/3
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|12x + 3y| < 15
What is the range of values for y, expressed in terms of x?
|12x + 3y| < 15
What is the range of values for y, expressed in terms of x?
Recall that with absolute values and "less than" inequalities, we have to hold the following:
12x + 3y < 15
AND
12x + 3y > –15
Otherwise written, this is:
–15 < 12x + 3y < 15
In this form, we can solve for y. First, we have to subtract x from all 3 parts of the inequality:
–15 – 12x < 3y < 15 – 12x
Now, we have to divide each element by 3:
(–15 – 12x)/3 < y < (15 – 12x)/3
This simplifies to:
–5 – 4x < y < 5 – 4x
Recall that with absolute values and "less than" inequalities, we have to hold the following:
12x + 3y < 15
AND
12x + 3y > –15
Otherwise written, this is:
–15 < 12x + 3y < 15
In this form, we can solve for y. First, we have to subtract x from all 3 parts of the inequality:
–15 – 12x < 3y < 15 – 12x
Now, we have to divide each element by 3:
(–15 – 12x)/3 < y < (15 – 12x)/3
This simplifies to:
–5 – 4x < y < 5 – 4x
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|4x + 14| > 30
What is a possible valid value of x?
|4x + 14| > 30
What is a possible valid value of x?
This inequality could be rewritten as:
4x + 14 > 30 OR 4x + 14 < –30
Solve each for x:
4x + 14 > 30; 4x > 16; x > 4
4x + 14 < –30; 4x < –44; x < –11
Therefore, anything between –11 and 4 (inclusive) will not work. Hence, the answer is 7.
This inequality could be rewritten as:
4x + 14 > 30 OR 4x + 14 < –30
Solve each for x:
4x + 14 > 30; 4x > 16; x > 4
4x + 14 < –30; 4x < –44; x < –11
Therefore, anything between –11 and 4 (inclusive) will not work. Hence, the answer is 7.
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If n and p are positive and 100_n_3_p_-1 = 25_n_, what is n-2 in terms of p ?
If n and p are positive and 100_n_3_p_-1 = 25_n_, what is n-2 in terms of p ?
To solve this problem, we look for an operation to perform on both sides that will leave n-2 by itself on one side. Dividing both sides by 25_n_-3 would leave n-2 by itself on the right side of the equqation, as shown below:
100n3p–1/25n–3 = 25n/25n–3
Remember that when dividing terms with the same base, we subtract the exponents, so the equation can be written as 100n0p–1/25 = n–2
Finally, we simplify to find 4_p–_1 = _n–_2.
To solve this problem, we look for an operation to perform on both sides that will leave n-2 by itself on one side. Dividing both sides by 25_n_-3 would leave n-2 by itself on the right side of the equqation, as shown below:
100n3p–1/25n–3 = 25n/25n–3
Remember that when dividing terms with the same base, we subtract the exponents, so the equation can be written as 100n0p–1/25 = n–2
Finally, we simplify to find 4_p–_1 = _n–_2.
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If 7y = 4x - 12, then x =
If 7y = 4x - 12, then x =
Adding 12 to both sides and dividing by 4 yields (7y+12)/4.
Adding 12 to both sides and dividing by 4 yields (7y+12)/4.
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Which of the statements describes the solution set for **–**7(x + 3) = **–**7x + 20 ?
Which of the statements describes the solution set for **–**7(x + 3) = **–**7x + 20 ?
By distribution we obtain **–**7x – 21 = – 7x + 20. This equation is never possibly true.
By distribution we obtain **–**7x – 21 = – 7x + 20. This equation is never possibly true.
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Given the inequality, |2_x_ – 2| > 20,
what is a possible value for x?
Given the inequality, |2_x_ – 2| > 20,
what is a possible value for x?
For this problem, we must take into account the absolute value.
First, we solve for 2_x_ – 2 > 20. But we must also solve for 2_x_ – 2 < –20 (please notice that we negate 20 and we also flip the inequality sign).
First step:
2_x_ – 2 > 20
2_x_ > 22
x > 11
Second step:
2_x_ – 2 < –20
2_x_ < –18
x < –9
Therefore, x > 11 and x < –9.
A possible value for x would be –10 since that is less than –9.
Note: the value 11 would not be a possible value for x because the inequality sign given does not include an equal sign.
For this problem, we must take into account the absolute value.
First, we solve for 2_x_ – 2 > 20. But we must also solve for 2_x_ – 2 < –20 (please notice that we negate 20 and we also flip the inequality sign).
First step:
2_x_ – 2 > 20
2_x_ > 22
x > 11
Second step:
2_x_ – 2 < –20
2_x_ < –18
x < –9
Therefore, x > 11 and x < –9.
A possible value for x would be –10 since that is less than –9.
Note: the value 11 would not be a possible value for x because the inequality sign given does not include an equal sign.
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Given the inequality above, which of the following MUST be true?
Given the inequality above, which of the following MUST be true?
Subtract
from both sides:


Subtract 7 from both sides:


Divide both sides by
:

Remember to switch the inequality when dividing by a negative number:

Since
is not an answer, we must find an answer that, at the very least, does not contradict the fact that
is less than (approximately) 4.67. Since any number that is less than 4.67 is also less than any number that is bigger than 4.67, we can be sure that
is less than 5.
Subtract
from both sides:
Subtract 7 from both sides:
Divide both sides by :
Remember to switch the inequality when dividing by a negative number:
Since is not an answer, we must find an answer that, at the very least, does not contradict the fact that
is less than (approximately) 4.67. Since any number that is less than 4.67 is also less than any number that is bigger than 4.67, we can be sure that
is less than 5.
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If 18 – w is 8 less than 32, what is the value of
?
If 18 – w is 8 less than 32, what is the value of ?
We need to rewrite this problem in mathematic terms.
8 less than 32 can be written as 32 – 8
so we will get the equation
18 – w = 32 – 8.
Now, we need to solve for w.
–w = 32 – 8 – 18
–w = 6
w = –6
Find the value of the given expression,
, by plugging in –6 for w.

We need to rewrite this problem in mathematic terms.
8 less than 32 can be written as 32 – 8
so we will get the equation
18 – w = 32 – 8.
Now, we need to solve for w.
–w = 32 – 8 – 18
–w = 6
w = –6
Find the value of the given expression, , by plugging in –6 for w.
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A function F is defined as follows:
for x2 > 1, F(x) = 4x2 + 2x – 2
for x2 < 1, F(x) = 4x2 – 2x + 2
What is the value of F(1/2)?
A function F is defined as follows:
for x2 > 1, F(x) = 4x2 + 2x – 2
for x2 < 1, F(x) = 4x2 – 2x + 2
What is the value of F(1/2)?
For F(1/2), x2=1/4, which is less than 1, so we use the bottom equation to solve. This gives F(1/2)= 4(1/2)2 – 2(1/2) + 2 = 1 – 1 + 2 = 2
For F(1/2), x2=1/4, which is less than 1, so we use the bottom equation to solve. This gives F(1/2)= 4(1/2)2 – 2(1/2) + 2 = 1 – 1 + 2 = 2
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If
, which of the following MUST be true?
I. 
II. 
III. 
If , which of the following MUST be true?
I.
II.
III.
Subtract 5 from both sides of the inequality:

Multiply both sides by 5:

Therefore only I must be true.
Subtract 5 from both sides of the inequality:
Multiply both sides by 5:
Therefore only I must be true.
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2x + 3y = 5a + 2b (1)
3x + 2y = 4a – b (2)
Express x2 – y2 in terms of a and b
2x + 3y = 5a + 2b (1)
3x + 2y = 4a – b (2)
Express x2 – y2 in terms of a and b
Add the two equations together to yield 5x + 5y = 9a + b, then factor out 5 to get 5(x + y) = 9a + b; divide both sides by 5 to get x + y = (9a + b)/5; subtract the two equations to get x - y = -a - 3b. So, x2 – y2 = (x + y)(x – y) = (9a + b)/5 (–a – 3b) = (–\[(9a)\]2 – 28ab – \[(3b)\]2)/5
Add the two equations together to yield 5x + 5y = 9a + b, then factor out 5 to get 5(x + y) = 9a + b; divide both sides by 5 to get x + y = (9a + b)/5; subtract the two equations to get x - y = -a - 3b. So, x2 – y2 = (x + y)(x – y) = (9a + b)/5 (–a – 3b) = (–\[(9a)\]2 – 28ab – \[(3b)\]2)/5
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If a positive integer a is divided by 7, the remainder is 4. What is the remainder if 3_a_ + 5 is divided by 3?
If a positive integer a is divided by 7, the remainder is 4. What is the remainder if 3_a_ + 5 is divided by 3?
The best way to solve this problem is to plug in an appropriate value for a. For example, plug-in 11 for a because 11 divided by 7 will give us a remainder of 4.
Then 3_a + 5_, where a = 11, gives us 38. Then 38 divided by 3 gives a remainder of 2.
The algebra method is as follows:
a divided by 7 gives us some positive integer b, with a remainder of 4.
Thus,
a / 7 = b 4/7
a / 7 = (7_b +_ 4) / 7
a = (7_b_ + 4)
then 3_a + 5 =_ 3 (7_b_ + 4) + 5
(3_a_+5)/3 = \[3(7_b_ + 4) + 5\] / 3
= (7_b_ + 4) + 5/3
The first half of this expression (7_b_ + 4) is a positive integer, but the second half of this expression (5/3) gives us a remainder of 2.
The best way to solve this problem is to plug in an appropriate value for a. For example, plug-in 11 for a because 11 divided by 7 will give us a remainder of 4.
Then 3_a + 5_, where a = 11, gives us 38. Then 38 divided by 3 gives a remainder of 2.
The algebra method is as follows:
a divided by 7 gives us some positive integer b, with a remainder of 4.
Thus,
a / 7 = b 4/7
a / 7 = (7_b +_ 4) / 7
a = (7_b_ + 4)
then 3_a + 5 =_ 3 (7_b_ + 4) + 5
(3_a_+5)/3 = \[3(7_b_ + 4) + 5\] / 3
= (7_b_ + 4) + 5/3
The first half of this expression (7_b_ + 4) is a positive integer, but the second half of this expression (5/3) gives us a remainder of 2.
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Simplify

Simplify
Divide the coefficients and subtract the exponents.
Divide the coefficients and subtract the exponents.
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What is a possible value for x in x2 – 12x + 36 = 0 ?
What is a possible value for x in x2 – 12x + 36 = 0 ?
You need to factor to find the possible values for x. You need to fill in the blanks with two numbers with a sum of -12 and a product of 36. In both sets of parenthesis, you know you will be subtracting since a negative times a negative is a positive and a negative plus a negative is a negative
(x –__)(x –__).
You should realize that 6 fits into both blanks.
You must now set each set of parenthesis equal to 0.
x – 6 = 0; x – 6 = 0
Solve both equations: x = 6
You need to factor to find the possible values for x. You need to fill in the blanks with two numbers with a sum of -12 and a product of 36. In both sets of parenthesis, you know you will be subtracting since a negative times a negative is a positive and a negative plus a negative is a negative
(x –__)(x –__).
You should realize that 6 fits into both blanks.
You must now set each set of parenthesis equal to 0.
x – 6 = 0; x – 6 = 0
Solve both equations: x = 6
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