### All PSAT Math Resources

## Example Questions

### Example Question #1 : How To Find Out When An Equation Has No Solution

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

y = (4x^{2} - 2)/(9 - x^{2})

**Possible Answers:**

no possible solution

3

0

6

**Correct answer:**

no possible solution

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.

### Example Question #1 : How To Find Out When An Equation Has No Solution

I. *x* = 0

II. *x* = –1

III. *x* = 1

**Possible Answers:**

III only

I, II, and III

I only

II only

II and III only

**Correct answer:**

I only

### Example Question #1 : How To Find Out When An Equation Has No Solution

**Possible Answers:**

1

–1/2

3

There is no solution

–3

**Correct answer:**

There is no solution

### Example Question #2 : How To Find Out When An Equation Has No Solution

**Possible Answers:**

None of the other answers

**Correct answer:**

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

### Example Question #1 : Linear / Rational / Variable Equations

Consider the equation

Which of the following is true?

**Possible Answers:**

The equation has exactly two solutions, which are of like sign.

The equation has exactly one solution, which is positive.

The equation has exactly one solution, which is negative.

The equation has exactly two solutions, which are of unlike sign.

The equation has no solution.

**Correct answer:**

The equation has exactly two solutions, which are of unlike sign.

Multiply the equation on both sides by LCM :

or

Substitution confirms that these are the solutions.

There are two solutions of unlike sign.

### Example Question #2 : Linear / Rational / Variable Equations

Which of the following equations has no solution?

**Possible Answers:**

Each of the equations in the other responses has no solution.

**Correct answer:**

Each of the equations in the other responses has no solution.

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.

### Example Question #3 : Linear / Rational / Variable Equations

Consider the equation

Which of the following is true?

**Possible Answers:**

The equation has exacty one real solution, which is negative.

The equation has exactly two real solutions, which are of like sign.

The equation has exactly two real solutions, which are of unlike sign.

The equation has exacty one real solution, which is positive.

The equation has no real solutions.

**Correct answer:**

The equation has exactly two real solutions, which are of unlike sign.

Multiply both sides by LCD :

or

There are two solutions of unlike sign.

### Example Question #4 : Linear / Rational / Variable Equations

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

**Possible Answers:**

**Correct 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.

### Example Question #1 : How To Find Out When An Equation Has No Solution

Which of the following equations has no real solutions?

**Possible Answers:**

Each of the equations given in the other choices has at least one real solution.

**Correct 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 .

### Example Question #6 : Linear / Rational / Variable Equations

Solve .

**Possible Answers:**

No solutions

**Correct answer:**

No solutions

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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