Is there any value of x that is consistent with the following?

Question

Is there any value of x that is consistent with the following?

"The value of the expression 3-2x is 5 more than twice the value of the expression."

Answer 1

So Lily S.,

Just transform that word problem into a math equation and try to solve it:

(3x - 2) = 5 + 2(3x - 2)

That works easily into:

-5 = (3x -2)

3x = -3

x = -1

So that value, and that value only!, is consistent with the statement.

If you had arrived at an equation: -4 = -4 for example, since x appears nowhere in it, x could be anything = infinite solutions.

But if you had arrived at an equation: -4 = 2 for example, that wouldn't ever be true, so it doesn't have a solution for x = no solutions.

There are certainly equations which have more than one, but still a finite number, of solutions. And then there are equations which have an infinite number of solutions, but they are discrete values.

You will learn about all these possibilities, eventually. And if you are so inclined, you may look for the beauty in the form of the equations.

For example, for a finite number of solutions, consisting of integers, consider expanding the factored equation series:

(x-1) = 0 (has solution x=1) : x-1 = 0

(x-2)(x-1) = 0 (has solutions x=1, 2) : x^2 - 3x + 2 = 0

(x-3)(x-2)(x-1) = 0 (has solutions x = 1, 2, 3) x^3 - 6x^2 + 11x - 6 = 0

Do you perceive an order yet? Probably not yet. But take it a little further and see what you find.

If this sort of problem catches your eye, you can easily set up and examine many other examples.

-- Cheers, --Mr. d.