ACT Math : Equations / Inequalities
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Example Questions
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?
25
17
10
7.5
25
First, simplify the equation so the absolute value is all that remains on the left side of the equation:
|2x – 25| = 10
Now create two equalities, one for 10 and one for –10.
2x – 25 = 10 and 2x – 25 = –10
2x = 35 and 2x = 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 x2 + 5x + 6.
2, 3
1, 4
2, –3
–2, 3
–2, –3
–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?
2
9
7
15
3
15
You set up the equation and you get: (x/3) + 2 = 2x – 23.
Add 23 to both sides: (x/3) + 25 = 2x
Multiply both sides by 3: x + 75 = 6x
Subtract x from both sides: 75 = 5x
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:
2x2 – 2x – 2 = 1 – x
First, we need to get everything on one side so that the equation equals zero.
2x2 - 2x -2 = 1-x
We need to add x to the left, and then subtract 1.
2x2 - 2x -2 +x - 1 = 0
2x2 - 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:
2x2 - x - 3 = 2x2 + 2x - 3x - 3 = 0
2x2 + 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 #1 : How To Find A Solution Set
If 3y = 2x – 7, then which of the following statements is correct?
they are equal
y is greater
not enough information given
x is greater
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 #1 : How To Find A Solution Set
|10 – 2| – |1 – 9| = ?
2
0
8
16
0
When taking the absolute value we realize that both absolute value operations yield 8, which gives us a difference of 0.
Example Question #4 : 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?
13
6
7
10
5
13
You set up the equation 5x – 23 = 3x + 3, then solve for x, giving you 13.
Example Question #1 : How To Find A Solution Set
What is the product of the two values of 
First, solve for the values of x by factoring.
Then, multiply the solutions to obtain the product.
Example Question #1 : How To Find A Solution Set
Solve for y:
Collecting terms leaves
And dividing by 
Example Question #2 : How To Find A Solution Set
Solve for x.
None of the other answers
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