Equations / Inequalities

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PSAT Math › Equations / Inequalities

Questions 1 - 10
1

If 5 + x is 5 more than 5,what is the value of 2_x_?

10

5

15

20

Explanation

5 more than 5 = 10

5 + x = 10

Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10

2

Which of the following is a root of the function f(x)=2x^2-7x-4 ?

x = -\frac {1}{2}

x = \frac{1}{2}

x = -4

x = -2

x = \frac{1}{4}

Explanation

The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.

f(x)=2x^2-7x-4 = 0

This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)

Because the coefficient in front of the x^2 is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.

2x^2-7x-4=2x^2-8x+x-4=0

We will then group the first two terms and the last two terms.

(2x^2-8x)+(x-4)=0

We will next factor out a 2_x_ from the first two terms.

(2x^2-8x)+(x-4)=2x(x-4)+1(x-4)=(2x+1)(x-4)=0

Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.

We now set each factor equal to zero and solve for x.

2x + 1 = 0

Subtract 1 from both sides.

2_x_ = –1

Divide both sides by 2.

x=-\frac{1}{2}

Now, we set x – 4 equal to 0.

x – 4 = 0

Add 4 to both sides.

x = 4

The roots of f(x) occur where x = -\frac{1}{2},4.

The answer is therefore x = -\frac {1}{2}.

3

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?

$500

$250

$750

$300

$625

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.

4

What property of arithmetic is demonstrated here?

Reflexive

Transitive

Commutative

Identity

Symmetric

Explanation

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

5

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?

$500

$250

$750

$300

$625

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.

6

Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?

2

3

4

6

7

Explanation

In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.

7

Factor the following equation.

x2 – 16

(x + 4)(x + 4)

(x – 4)(x – 4)

(x + 4)(x – 4)

(x)(x – 4)

(x2)(4 – 2)

Explanation

The correct answer is (x + 4)(x – 4)

We neen to factor x2 – 16 to solve. We know that each parenthesis will contain an x to make the x2. We know that the root of 16 is 4 and since it is negative and no value of x is present we can tell that one 4 must be positive and the other negative. If we work it from the multiple choice answers we will see that when multiplying it out we get x2 + 4x – 4x – 16. 4x – 4x cancels out and we are left with our answer.

8

Factor the following equation.

x2 – 16

(x + 4)(x + 4)

(x – 4)(x – 4)

(x + 4)(x – 4)

(x)(x – 4)

(x2)(4 – 2)

Explanation

The correct answer is (x + 4)(x – 4)

We neen to factor x2 – 16 to solve. We know that each parenthesis will contain an x to make the x2. We know that the root of 16 is 4 and since it is negative and no value of x is present we can tell that one 4 must be positive and the other negative. If we work it from the multiple choice answers we will see that when multiplying it out we get x2 + 4x – 4x – 16. 4x – 4x cancels out and we are left with our answer.

9

Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?

2

3

4

6

7

Explanation

In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.

10

Which of the following is a root of the function f(x)=2x^2-7x-4 ?

x = -\frac {1}{2}

x = \frac{1}{2}

x = -4

x = -2

x = \frac{1}{4}

Explanation

The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.

f(x)=2x^2-7x-4 = 0

This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)

Because the coefficient in front of the x^2 is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.

2x^2-7x-4=2x^2-8x+x-4=0

We will then group the first two terms and the last two terms.

(2x^2-8x)+(x-4)=0

We will next factor out a 2_x_ from the first two terms.

(2x^2-8x)+(x-4)=2x(x-4)+1(x-4)=(2x+1)(x-4)=0

Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.

We now set each factor equal to zero and solve for x.

2x + 1 = 0

Subtract 1 from both sides.

2_x_ = –1

Divide both sides by 2.

x=-\frac{1}{2}

Now, we set x – 4 equal to 0.

x – 4 = 0

Add 4 to both sides.

x = 4

The roots of f(x) occur where x = -\frac{1}{2},4.

The answer is therefore x = -\frac {1}{2}.

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