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PSAT Math : Algebra

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Example Questions

Example Question #2 : How To Find The Solution To An Inequality With Division

Solve the inequality

$\frac{3x}{5} > 6$

Possible Answers:

x > 9

x < 10

x > 3

x > 10

Correct answer:

x > 10

Explanation:

First, multiplying each side of the equality by gives 3x > 30 . Next, dividing each side of the inequality by will solve for ; x > 10 .

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Example Question #1 : How To Find The Solution To An Inequality With Division

What is the solution set of the inequality $\dpi{100} \small 3x+8<35$ ?

Possible Answers:

$\dpi{100} \small x>9$

$\dpi{100} \small x>27$

$\dpi{100} \small x<35$

$\dpi{100} \small x<27$

$\dpi{100} \small x<9$

Correct answer:

$\dpi{100} \small x<9$

Explanation:

We simplify this inequality similarly to how we would simplify an equation

$\dpi{100} \small 3x+8-8<35-8$

$\dpi{100} \small \frac{3x}{3}<\frac{27}{3}$

Thus $\dpi{100} \small x<9$

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Example Question #4 : Inequalities

What is a solution set of the inequality 2x+12>42 ?

Possible Answers:

x>4

x<9

x<15

$x>\frac{3}{2}$

x>15

Correct answer:

x>15

Explanation:

In order to find the solution set, we solve 2x+12>42 as we would an equation:

2x+12>42

2x>30

x>15

Therefore, the solution set is any value of x>15 .

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Example Question #1 : Linear / Rational / Variable Equations

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 #2 : Equations / Inequalities

Undefined_denom3

I. x = 0

II. x = –1

III. x = 1

Possible Answers:

I only

I, II, and III

II only

III only

II and III only

Correct answer:

I only

Explanation:

Undefined_denom2

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Example Question #1 : Linear / Rational / Variable Equations

Nosol1

Possible Answers:

–1/2

There is no solution

–3

Correct answer:

There is no solution

Explanation:

Nosol2

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Example Question #2 : Linear / Rational / Variable Equations

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

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

Possible Answers:

$\small 4$

$\small -4$

$\small 0$

None of the other answers

$\small 28$

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 #3 : 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 two solutions, which are of unlike sign.

The equation has exactly one solution, which is negative.

The equation has exactly one solution, which is positive.

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

The equation has no solution.

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 + 2 | = 8

6- |x + 6 | = 8

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

6- |x + 8 | = 8

6- |x | = 8

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 #3 : 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 exactly two real solutions, which are of unlike sign.

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

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

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

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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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #2 : How To Find The Solution To An Inequality With Division

Example Question #1 : How To Find The Solution To An Inequality With Division

Example Question #4 : Inequalities

Example Question #1 : Linear / Rational / Variable Equations

Example Question #2 : Equations / Inequalities

Example Question #1 : Linear / Rational / Variable Equations

Example Question #2 : Linear / Rational / Variable Equations

Example Question #3 : Linear / Rational / Variable Equations

Example Question #2 : Linear / Rational / Variable Equations

Example Question #3 : Linear / Rational / Variable Equations

All PSAT Math Resources

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