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PSAT Math : Linear / Rational / Variable Equations

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Example Question #3 : How To Find Out When An Equation Has No Solution

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

y = (4x² - 2)/(9 - x²)

Possible Answers:

no possible solution

Correct answer:

no possible solution

Explanation:

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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Example Question #1 : Linear / Rational / Variable Equations

Undefined_denom3

I. x = 0

II. x = –1

III. x = 1

Possible Answers:

III only

I, II, and III

II and III only

I only

II only

Correct answer:

I only

Explanation:

Undefined_denom2

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Example Question #5 : How To Find Out When An Equation Has No Solution

Nosol1

Possible Answers:

–1/2

–3

There is no solution

Correct answer:

There is no solution

Explanation:

Nosol2

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Example Question #6 : How To Find Out When An Equation Has No Solution

$\small h\left ( x\right )=\frac{28}{x+4}$

$\small \textup{For which of the following values of}\,x\,\textup{is the above function undefined?}$

Possible Answers:

None of the other answers

$\small 4$

$\small 0$

$\small 28$

$\small -4$

Correct answer:

$\small -4$

Explanation:

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

$\small x+4=0$

$\small x=-4$

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Example Question #1 : Linear / Rational / Variable Equations

Consider the equation

$\frac{x+9}{x}+\frac{x+3}{x-2} = \frac{4x+11}{x}$

Which of the following is true?

Possible Answers:

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.

The equation has no solution.

The equation has exactly one solution, which is negative.

Correct answer:

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

Explanation:

Multiply the equation on both sides by LCM $\frac{x(x-2)}{1}$ :

$\frac{x+9}{x}+\frac{x+3}{x-2} = \frac{4x+11}{x}$

$\frac{x(x-2)}{1} \cdot \frac{x+9}{x}+\frac{x(x-2)}{1} \cdot\frac{x+3}{x-2} = \frac{x(x-2)}{1} \cdot\frac{4x+11}{x}$

$(x-2) \left ( x+9 \right ) + x( x+3 )= (x-2) (4x+11)$

$\left ( x^{2}+7x -18\right ) + ( x^{2}+3x )= 4x^{2}+3x -22$

$4x^{2}+3x -22 = x^{2}+7x -18 + x^{2}+3x$

$4x^{2}+3x -22 = 2x^{2}+10x -18$

$4x^{2}+3x -22 - 2x^{2}-10x +18 = 2x^{2}+10x -18 - 2x^{2}-10x +18$

$2x^{2}-7x -4 = 0$

(x-4)(2x+1)= 0

x-4 = 0

x= 4

2x+1 = 0

2x = -1

$x = -\frac{1}{2}$

Substitution confirms that these are the solutions.

There are two solutions of unlike sign.

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Example Question #2 : Linear / Rational / Variable Equations

Which of the following equations has no solution?

Possible Answers:

6- |x + 8 | = 8

6- |x | = 8

6- |x + 6 | = 8

6- |x + 2 | = 8

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

Correct answer:

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

Explanation:

The problem is basically asking for what value of the equation

6- |x + A | = 8

has no solution.

We can simplify as folllows:

6- |x + A | - 6 = 8 - 6

- |x + A | = 2

|x + A | = - 2

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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Example Question #1 : Linear / Rational / Variable Equations

Consider the equation

$\frac{4}{y+3} = \frac{y+6}{10}$

Which of the following is true?

Possible Answers:

The equation has no real solutions.

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

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

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

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

Correct answer:

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

Explanation:

Multiply both sides by LCD $\frac{10(y+3)}{1}$ :

$\frac{4}{y+3} = \frac{y+6}{10}$

$\frac{4}{y+3}\cdot \frac{10(y+3)}{1} = \frac{y+6}{10} \cdot \frac{10(y+3)}{1}$

$4 \cdot 10 = (y+6) (y+3)$

$y^{2}+9y+18 = 40$

$y^{2}+9y+18- 40 = 40 - 40$

$y^{2}+9y-22 = 0$

(y+11)(y-2) = 0

$y+11 \Rightarrow y = -11$

$y-2 = 0 \Rightarrow y = 2$

There are two solutions of unlike sign.

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

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

Possible Answers:

4x + 3(x-10) = 2x + 5 (x - 8)

4x + 3(x-10) = 2x + 5 (x - 6)

4x + 3(x-10) = 2x + 5 (x - 12)

4x + 3(x-10) = 2x + 5 (x - 10)

4x + 3(x-10) = 2x + 5 (x - 4)

Correct answer:

4x + 3(x-10) = 2x + 5 (x - 6)

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

4x + 3(x-10) = 2x + 5 (x - A)

have a solution set other than the empty set?

We can simplify as follows:

4x + 3x-30= 2x + 5 x - 5A

7x-30=7x - 5A

-30 = -5A

If -5A and are not equivalent expressions, the solution set is the empty set. If -5A and are equivalent expressions, the solution set is the set of all real numbers; this happens if and only if:

-5A = -30

$-5A \div (-5)= -30 \div (-5)$

A = 6

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

4x + 3(x-10) = 2x + 5 (x - 6)

which is the correct choice.

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Example Question #5 : Linear / Rational / Variable Equations

Which of the following equations has no real solutions?

Possible Answers:

$-14 - \sqrt[4]{x - 7 } = 7$

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

$14 - \sqrt[4]{x - 7 } = -7$

$-14 - \sqrt[3]{x - 7 } = 7$

$14 - \sqrt[3]{x - 7 } = -7$

Correct answer:

$-14 - \sqrt[4]{x - 7 } = 7$

Explanation:

We can examine each individually.

$14 - \sqrt[3]{x - 7 } = -7$

$14 - \sqrt[3]{x - 7 } + 7 + \sqrt[3]{x - 7 } = -7 + 7 + \sqrt[3]{x - 7 }$

$21 = \sqrt[3]{x - 7 }$

$21^{3} =\left ( \sqrt[3]{x - 7 } \right )^{3}$

x-7 = 9,261

x = 9,268

This equation has a solution.

$-14 - \sqrt[3]{x - 7 } = 7$

$-14 - \sqrt[3]{x - 7 } - 7 + \sqrt[3]{x - 7 } = 7 - 7 + \sqrt[3]{x - 7 }$

$-21 = \sqrt[3]{x - 7 }$

$\left (-21 \right )^{3} =\left ( \sqrt[3]{x - 7 } \right )^{3}$

x-7 =- 9,261

x =- 9,254

This equation has a solution.

$14 - \sqrt[4]{x - 7 } = -7$

$14 - \sqrt[4]{x - 7 } + 7 + \sqrt[4]{x - 7 } = -7 + 7 + \sqrt[4]{x - 7 }$

$21 = \sqrt[4]{x - 7 }$

$21^{4} =\left ( \sqrt[4]{x - 7 } \right )^{4}$

x-7 = 194,481

x = 194,488

This equation has a solution.

$-14 - \sqrt[4]{x - 7 } = 7$

$-14 - \sqrt[4]{x - 7 } - 7 + \sqrt[4]{x - 7 } = 7 - 7 + \sqrt[4]{x - 7 }$

$-21 = \sqrt[4]{x - 7 }$

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

The correct choice is $-14 - \sqrt[4]{x - 7 } = 7$ .

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Example Question #1 : Equations / Inequalities

Solve $\left | 3-4x\right |<0$ .

Possible Answers:

$x<\frac{4}{3}$

No solutions

$x>\frac{4}{3}$

$x<\frac{3}{4}$

$x>\frac{3}{4}$

Correct answer:

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.

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PSAT Math : Linear / Rational / Variable Equations

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #3 : How To Find Out When An Equation Has No Solution

Example Question #1 : Linear / Rational / Variable Equations

Example Question #5 : How To Find Out When An Equation Has No Solution

Example Question #6 : How To Find Out When An Equation Has No Solution

Example Question #1 : Linear / Rational / Variable Equations

Example Question #2 : Linear / Rational / Variable Equations

Example Question #1 : Linear / Rational / Variable Equations

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

Example Question #5 : Linear / Rational / Variable Equations

Example Question #1 : Equations / Inequalities

All PSAT Math Resources

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