# PSAT Math : Equations / Inequalities

## Example Questions

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

Consider the equation

Which of the following is true?

The equation has exactly one solution, which is negative.

The equation has no solution.

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

The equation has exactly one solution, which is positive.

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

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

Explanation:

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

Which of the following equations has no solution?

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

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

Explanation:

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?

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

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

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

The equation has no real solutions.

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

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

Explanation:

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?

Explanation:

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 #5 : Linear / Rational / Variable Equations

Which of the following equations has no real solutions?

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

Explanation:

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

Solve .

No solutions

No solutions

Explanation:

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

–2

1

0

–1

2

2

Explanation:

b/(m– 1)

–b/(+ 1)

bm/(m+ 1)

–bm/(m+ 1)

b/(m+ 1)

b/(m+ 1)

Explanation:

### Example Question #3 : How To Find The Solution To A Rational Equation With Lcd

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

Explanation:

### Example Question #1 : Equations / Inequalities

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

2/7

1/7

1/6

7

5