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  1a) x = 4

  1c) x = 6

  1d) x = 10

  1f) x = -42

Step-by-step explanation:

1a) 3x -8 = 4

This is a "two step" equation with a variable term and constant on the same side of the equal sign. Step 1, add the opposite of that constant to both sides of the equation:

  3x = 12

Step 2: divide both sides of the equation by the coefficient of the variable:

  x = 4

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1c) This is similar to 1a, except that it has variables on both sides of the equal sign. Find the variable term with the smallest coefficient. Here, it is -x, because the coefficient -1 is less than 2. Add the opposite of that term to both sides of the equation:

  3x -8 = 10

Now, you have a 2-step equation in the same form as 1a. Solve it the same way.

  3x = 18 . . . . add 8

  x = 6 . . . . . . divide by 3

Check

  2(6) -8 = 10 -(6)

  12 -8 = 10 -6

  4 = 4 . . . . . true; answer is correct

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1d) After you eliminate parentheses using the distributive property, you end up with a 3-step equation as in 1c.

  6x -12 = 3x +2x -2

  6x -12 = 5x -2 . . . . . . combine like terms. (5x is the smallest x-term)

  x -12 = -2 . . . . . . . . . subtract 5x

  x = 10 . . . . . . . . . . . . add 12

Check

  6(10 -2) = 3(10) +2(10 -1)

  6(8) = 30 +2(9)

  48 = 30 +18

  48 = 48 . . . . true; answer is correct

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1f) You can work this as is using the fractional coefficients. Most folks prefer to "clear fractions" first. To do that, you need to find the least common denominator and multiply both sides of the equation by it. The least common denominator of 1/3 and 1/2 is 6.

  2x = 3x +42 . . . . . multiply both sides by 6

  0 = x + 42 . . . . . . . subtract the smallest x-term from both sides

  -42 = x . . . . . . . . . add -42

Check

  (-42)/3 = (-42)/2 +7

  -14 = -21 +7

  -14 = -14 . . . . . . . true; answer is correct

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If you work this using the given coefficients, you can subtract 1/3x to get ...

  0 = x/6 +7

  -7 = x/6 . . . . . subtract 7

  -42 = x . . . . . . multiply by 6

You notice that we had to multiply by 6 in the solution process anyway. Doing that first makes the fractions go away, so is often the preferred solution method.