### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find A Solution Set

What is the product of the smallest and largest integers that satisfy the equation |2x – 4|<6?

**Possible Answers:**

5

0

–5

–10

**Correct answer:**

0

Solve the equations

(2x – 4) < 6 AND (2x – 4) > –6

x < 5 AND x > –1

The largest integer that satisfies this equation is 0, and the smallest is 4. Their product is 0.

### Example Question #1 : How To Find A Solution Set

|2x – 25| – 3 = 7. There are two solutions to this problem. What is the sum of those solutions?

**Possible Answers:**

17

10

7.5

25

**Correct answer:**

25

First, simplify the equation so the absolute value is all that remains on the left side of the equation:

|2*x* – 25| = 10

Now create two equalities, one for 10 and one for –10.

2*x* – 25 = 10 and 2*x* – 25 = –10

2*x* = 35 and 2*x* = 15

*x* = 17.5 and *x* = 7.5

The two solutions are 7.5 and 17.5. 17.5 + 7.5 = 25

### Example Question #1 : How To Find A Solution Set

Find the roots of the equation *x*^{2 }+ 5*x* + 6.

**Possible Answers:**

–2, –3

1, 4

2, –3

2, 3

–2, 3

**Correct answer:**

–2, –3

Factoring gives us (*x* + 2)(*x* + 3). This yields *x* = –2, –3.

### Example Question #1 : How To Find A Solution Set

When you divide a number by 3 and then add 2, the result is the same as when you multiply the same number by 2 then subtract 23. What is the number?

**Possible Answers:**

9

3

7

2

15

**Correct answer:**

15

You set up the equation and you get: (*x*/3) + 2 = 2*x –* 23.

Add 23 to both sides: (*x*/3) + 25 = 2*x*

Multiply both sides by 3: *x* + 75 = 6*x*

Subtract *x* from both sides: 75 = 5*x*

Divide by 5 and get *x *= 15

### Example Question #1 : How To Find A Solution Set

Find the sum of the solutions to the equation:

2x^{2 } **– **2x **– **2 = 1 **– **x

**Possible Answers:**

**Correct answer:**

First, we need to get everything on one side so that the equation equals zero.

2x^{2 }- 2x -2 = 1-x

We need to add x to the left, and then subtract 1.

2x^{2 }- 2x -2 +x - 1 = 0

2x^{2 }- x - 3 = 0

Now we need to factor the binomial. In order to do this, we need to multiply the outer two coefficients, which will give us 2(-3) = -6. We need to find two numbers that will mutiply to give us -6. We also need these two numers to equal -1 when we add them, because -1 is the coefficient of the x term.

If we use +2 and -3, then these two numbers will multiply to give us -6 and add to give us -1. Now we can rewrite the equation as follows:

2x^{2 }- x - 3 = 2x^{2 }+ 2x - 3x - 3 = 0

2x^{2 }+ 2x - 3x - 3 = 0

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

2x(x+1) -3(x+1) = 0

(2x-3)(x+1) = 0

This means that either 2x - 3 = 0, or x + 1 = 0. So the values of x that solve the equation are 3/2 and -1.

The question asks us for the sum of the solutions, so we must add 3/2 and -1, which would give us 1/2.

### Example Question #2 : How To Find A Solution Set

If 3y = 2x – 7, then which of the following statements is correct?

**Possible Answers:**

x is greater

not enough information given

they are equal

y is greater

**Correct answer:**

not enough information given

If we set one variable to the other we would get y = (2x – 7)/3 or x = (3y + 7)/2, but we aren't given any clues to what the values of x and y are and we can assume they could be any number. If x = 7/2, then y = 0. If y = -7/3, then x = 0. Let's try some other numbers. If y = –10, then x = –37/2. So for the first two examples, x is greater than y. In the last example, y is greater than x. We need more information to determine whether x or y is greater. The correct answer is not enough information given.

### Example Question #3 : How To Find A Solution Set

|10 **–** 2| – |1 – 9| = ?

**Possible Answers:**

16

8

2

0

**Correct answer:**

0

When taking the absolute value we realize that both absolute value operations yield 8, which gives us a difference of 0.

### Example Question #1 : How To Find A Solution Set

When you multiply a number by 5 and then subtract 23, the result is the same as when you multiplied the same number by 3 then added 3. What is the number?

**Possible Answers:**

13

6

7

5

10

**Correct answer:**

13

You set up the equation 5x – 23 = 3x + 3, then solve for x, giving you 13.

### Example Question #3 : How To Find A Solution Set

What is the product of the two values of that satisfy the following equation?

**Possible Answers:**

**Correct answer:**

First, solve for the values of *x* by factoring.

or

Then, multiply the solutions to obtain the product.

### Example Question #1 : How To Find A Solution Set

Solve for y:

**Possible Answers:**

**Correct answer:**

Collecting terms leaves

And dividing by yields

### All ACT Math Resources

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