PSAT Math : Equations / Inequalities

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #45 : Equations / Inequalities

Pets Plus makes bird houses.  Their monthly fixed expenses are $750.  The cost for each bird house is $15.  The bird houses sell for $40.

What is the monthly break-even point at Pets Plus?

Possible Answers:

40

35

30

50

25

Correct answer:

30

Explanation:

Let x= the number of bird houses sold each month.

Revenue = 40x

Costs=15x+750

The break-even point is where the revenue is the same as the costs:

Revenue=Costs 

40x=15x+750 

Solve for x:

x=30

Therefore, Pets Plus must sell 30 bird houses to break-even.

Example Question #51 : Algebra

Pets Plus makes bird houses.  Their monthly fixed expenses are $750.  The cost for each bird house is $15.  The bird houses sell for $40.

If Pets Plus sells 50 bird houses, what is the profit?

Possible Answers:

\$300

\$750

\$500

\$625

\$250

Correct answer:

\$500

Explanation:

Let x = the number of birdhouses sold each month.

Revenue=40x

Costs=15x+750

Profit = Revenue-Costs 

=40x-15x-750 

=25x-750

Substituting in 50 for x gives an answer of 500, so the profit on 50 birdhouses is $500.

Example Question #52 : Algebra

George is three times older than Joey.  The sum of their ages is 16.  What is the product of their ages?

Possible Answers:

Correct answer:

Explanation:

Let  = Joey's age and  = George's age.

Then the equation to solve becomes .

Therefore, Joey is 4 years old and George is 12 years old, so the product of their ages is 48.

Example Question #53 : Algebra

Three consecutive even numbers add to 42.  What is the middle number?

Possible Answers:

10

14

16

18

12

Correct answer:

14

Explanation:

Let x = 1st even number, x+2 = 2nd even number, and x+4 = 3rd even number.

Then the equation to solve becomes x+(x+2)+(x+4)=42.

3x+6=42

Thus x=12,x+2=14,\ and\ x+4=16, so the middle number is 14.

Example Question #54 : Algebra

Consider the following equation:

Which of the following must be true?

Possible Answers:

Correct answer:

Explanation:

The quantity inside the absolute value brackets must equal either  or , depending on whether the quantity inside the brackets is positive or negative. We therefore have two seperate equations:

To solve the first equation, add 9 to both sides:

Subtract  from both sides:

This is the first solution. Now let's look at the second equation. The distributive law gives us:

Add 9 to both sides:

Add  to both sides:

Divide both sides by 3:

Therefore, is either 4 or 6. 

Statement  does NOT have to be true because  can also equal 4.

Statement  must be true because both 4 and 6 are positive .

Finally, statement  always holds because 4 and 6 are both even. 

Example Question #882 : Psat Mathematics

What property of arithmetic is demonstrated here? 

Possible Answers:

Commutative

Reflexive

Symmetric

Identity

Transitive

Correct answer:

Reflexive

Explanation:

The statement expresses the idea that any number is equal to itself. This is the reflexive property of equality.

Example Question #1 : How To Find The Solution To A Quadratic Equation

If f(x) = -x2 + 6x - 5, then which could be the value of a if f(a) = f(1.5)?

Possible Answers:
1
4
4.5
3.5
2.5
Correct answer: 4.5
Explanation:

We need to input 1.5 into our function, then we need to input "a" into our function and set these results equal.

f(a) = f(1.5)

f(a) = -(1.5)2 +6(1.5) -5

f(a) = -2.25 + 9 - 5

f(a) = 1.75

-a2 + 6a -5 = 1.75

Multiply both sides by 4, so that we can work with only whole numbers coefficients.

-4a2 + 24a - 20 = 7

Subtract 7 from both sides.

-4a2 + 24a - 27 = 0

Multiply both sides by negative one, just to make more positive coefficients, which are usually easier to work with.

4a2 - 24a + 27 = 0

In order to factor this, we need to mutiply the outer coefficients, which gives us 4(27) = 108. We need to think of two numbers that multiply to give us 108, but add to give us -24. These two numbers are -6 and -18. Now we rewrite the equation as:

4a2 - 6a -18a + 27 = 0

We can now group the first two terms and the last two terms, and then we can factor.

(4a2 - 6a )+(-18a + 27) = 0

2a(2a-3) + -9(2a - 3) = 0

(2a-9)(2a-3) = 0

This means that 2a - 9 =0, or 2a - 3 = 0.

2a - 9 = 0

2a = 9

a = 9/2 = 4.5

2a - 3 = 0

a = 3/2 = 1.5

So a can be either 1.5 or 4.5.

The only answer choice available that could be a is 4.5.

Example Question #11 : Quadratic Equations

Solve for x:  2(x + 1)2 – 5 = 27

Possible Answers:

–2 or 5

3 or 4

–2 or 4

–3 or 2

3 or –5

Correct answer:

3 or –5

Explanation:

Quadratic equations generally have two answers.  We add 5 to both sides and then divide by 2 to get the quadratic expression on one side of the equation: (x + 1)2 = 16.   By taking the square root of both sides we get x + 1 = –4 or x + 1 = 4.  Then we subtract 1 from both sides to get x = –5 or x = 3.

Example Question #11 : Quadratic Equations

Two consecutive positive multiples of three have a product of 54. What is the sum of the two numbers?

Possible Answers:

9

15

6

12

3

Correct answer:

15

Explanation:

Define the variables to be x = first multiple of three and x + 3 = the next consecutive multiple of 3.

Knowing the product of these two numbers is 54 we get the equation x(x + 3) = 54. To solve this quadratic equation we need to multiply it out and set it to zero then factor it. So x2 + 3x – 54 = 0 becomes (x + 9)(x – 6) = 0.  Solving for x we get x = –9 or x = 6 and only the positive number is correct.  So the two numbers are 6 and 9 and their sum is 15.

Example Question #3 : How To Find The Solution To A Quadratic Equation

Solve 3x2 + 10x = –3

Possible Answers:

x = –1/9 or –9

x = –1/6 or –6

x = –2/3 or –2

x = –4/3 or –1

x = –1/3 or –3

Correct answer:

x = –1/3 or –3

Explanation:

Generally, quadratic equations have two answers.

First, the equations must be put in standard form: 3x2 + 10x + 3 = 0

Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.

Third, check the answer by plugging the answers back into the original equation.

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