Solution Sets - ACT Math

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

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

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.

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.

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.

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.

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

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

Collecting terms leaves

And dividing by yields

or

First, solve for the values of x by factoring.

or

Then, multiply the solutions to obtain the product.