Question

PLEASE HELP.... Find the solution to the following equations below and identify either one solution, no solution, or infinite solutions. (Also if you can, what does it mean to have no solution or Infinite solutions? I don't need to know this if you don't know)
A. 3 (x + 4) = 3x + 11

B. -2 (x + 3) = -2x - 6

C. 4 (x - 1) = 1/2 (x - 8)

D. 3x - 7 = 4 + 6 = 4x

Answer 1

When an equation has no solution, it means that no x can make the equation be a true statement. The opposite is when an equation has infinitely many solutions, it means everything put inf or x makes it true. One solution lies in between, and there is exactly one x that makes it true.

Let's look at the equations and see which are which.

A--

3 (x + 4) = 3x + 11

3x + 12 = 3x + 11 by the distributive property on the left side

12 = 11 by subtracting 3x on both sides.

Notice here that 12 = 11. That's always false. This kind of always false statement, rather than x = _____, tells us we have no solutions.

B---

-2 (x + 3) = -2x - 6

-2x - 6 = -2x -6

-6 = -6 by similar steps as in A

This equation gives something that is always true, that -6 = -6. When you get the same thing on both sides, this is a clue that you have infinitely many solutions.

C---

4 (x - 1) = 1/2 (x - 8)

4x - 4 = 1/2x - 4 by distributing on both sides

8x - 8 = x - 8 by multiplying the whole equation by 2 to remove fractions.

7x - 8 = -8 by subtracting x from both sides

7x = 0 by adding 8 to both sides

x = 0 by dividing 7 on both sides.

Here, we have exactly one solution, zero.

D---

This equation should be checked given the presence of a double equals sign.

Thus, we have the following.

A - No solution

B - Infinitely many solutions

C - Exactly one solution at x = 0

D - Check the double equals to see if it's a typo.