All ACT Math Resources
Example Questions
Example Question #2 : 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?
7
15
9
2
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 #3 : 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 #3 : How To Find A Solution Set
If 3y = 2x – 7, then which of the following statements is correct?
they are equal
not enough information given
x is greater
y 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 #2 : Solution Sets
|10 – 2| – |1 – 9| = ?
8
0
2
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 #2 : 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?
6
5
7
13
10
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 that satisfy the following equation?
First, solve for the values of x by factoring.
or
Then, multiply the solutions to obtain the product.
Example Question #2 : How To Find A Solution Set
Solve for y:
Collecting terms leaves
And dividing by yields
Example Question #2 : How To Find A Solution Set
Solve for x.
None of the other answers
or
Example Question #471 : Algebra
Given the follow inequality, which of the following presents a range of possible answers for the inequality: –3 < 3x + 2 ≤ 3.5
(–3, 1/2)
(–1,1)
( –2, 2)
(–1, ½)
(½, 1)
(–1, ½)
If you plug in the outer limits of the given ranges, (–1, ½) is the only combination that fits within the given equation. It is important to remember that "<" means “less than,” and "≤" means “less than or equal.” For example, if you answered (–2,2), plugging in 2 would make the the expression equal 8, which is greater than 3.5. And plugging in –2 for x would make the expression equal –4, which is less than –3, not greater. However, plugging in the correct answer (–1, ½) gives you –1 as your lower limit and 3.5 as your upper limit, which satisfies the equation. Both limits of the data set must satisfy the equation.
Example Question #12 : Solution Sets
If , what is the product of the largest and smallest integers that satisfy the inequality?
0
–10
–5
5
7
0
The inequality in the question possesses an absolute value; therefore, we most solve for the variable being less than positive 6 and greater than negative 6. Let's start with the positive solution.
Add 4 to both sides of the inequality.
Divide both sides of the inequality by 2.
Now, let's solve for the negative solution
Add 4 to both sides of the inequality.
Divide both sides of the inequality by 2.
Using these solutions we can write the following statement:
The smallest integer that satisfies this equation is 0, and the largest is 4. Their product is 0.