ACT Math Help » Algebra » Equations / Inequalities » Solution Sets » How to find a solution set

Explanation:

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.

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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

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?

Explanation:

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

Explanation:

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

2×2 – 2x -2 = 1-x

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

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2×2 – 2x -2 +x – 1 = 0

2×2 – 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:

2×2 – x – 3 = 2×2 + 2x – 3x – 3 = 0

2×2 + 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.

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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.