Linear / Rational / Variable Equations - PSAT Math

Card 0 of 490

Question

Consider the equation

Which of the following is true?

Answer

Multiply both sides by LCD :

or

There are two solutions of unlike sign.

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Question

A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?

Answer

This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.

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Question

The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?

Answer

Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.

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Question

Given f(x) = x2 – 9. What are the zeroes of the function?

Answer

The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.

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Question

Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?

Answer

In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.

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Question

Find the solution to the following equation if x = 3:

y = (4x2 - 2)/(9 - x2)

Answer

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

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Question

Undefined_denom3

I. x = 0

II. x = –1

III. x = 1

Answer

Undefined_denom2

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Question

Nosol1

Answer

Nosol2

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Question

Answer

A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.

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Question

Consider the equation

Which of the following is true?

Answer

Multiply the equation on both sides by LCM :

or

Substitution confirms that these are the solutions.

There are two solutions of unlike sign.

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Question

All of the following equations have no solution except for which one?

Answer

Since all of the equations have the same symbols save for one number, the problem is essentially as follows:

For what value of does the equation

have a solution set other than the empty set?

We can simplify as follows:

If and are not equivalent expressions, the solution set is the empty set. If and are equivalent expressions, the solution set is the set of all real numbers; this happens if and only if:

Therefore, the only equation among the given choices whose solution set is not the empty set is the equation

which is the correct choice.

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Question

Which of the following equations has no solution?

Answer

The problem is basically asking for what value of the equation

has no solution.

We can simplify as folllows:

Since the absolute value of a number must be nonnegative, regardless of the value of , this equation can never have a solution. Therefore, the correct response is that none of the given equations has a solution.

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Question

Which of the following equations has no real solutions?

Answer

We can examine each individually.

This equation has a solution.

This equation has a solution.

This equation has a solution.

This equation has no solution, since a fourth root of a number must be nonnegative.

The correct choice is .

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Question

Solve .

Answer

By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.

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Question

Rational_5

Answer

Rational_2

Rational_3

Rational_4

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Question

In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

Answer

Pkm_7-21-13

Pkm2_7-21-13

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Question

Linesmb1

Answer

Linesmb5

Linesmb4

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Question

If 6_x_ = 42 and xk = 2, what is the value of k?

Answer

Solve the first equation for x by dividing both sides of the equation by 6; the result is 7. Solve the second equation for k by dividing both sides of the equation by x, which we now know is 7. The result is 2/7.

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Question

If 4_x_ + 5 = 13_x_ + 4 – x – 9, then x = ?

Answer

Start by combining like terms.

4_x_ + 5 = 13_x_ + 4 – x – 9

4_x_ + 5 = 12_x_ – 5

–8_x_ = –10

x = 5/4

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Question

If 3 – 3_x_ < 20, which of the following could not be a value of x?

Answer

First we solve for x.

Subtracting 3 from both sides gives us –3_x_ < 17.

Dividing by –3 gives us x > –17/3.

–6 is less than –17/3.

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