Question

How many solutions exist for the given equation?

1/2(x+ 12) = 4x-1
zero
one
two
O infinitely many

Answer 1

one

Explanation

For a polynomial function, the number of solutions, or zeros, is generally equivalent to the highest degree. For example, a cubic function (e.g.
`x^3+2x^2`) has 3 real zeros (x=-2, x=0, and x=0) and a quadratic (e.g.
`x^2-1`) has 2 real zeros (x=-1 and x=1).

For your problem,
`(1)/(2) (x+12)=4x-1`, the highest degree is 1 since this is a linear function. To prove there is only one solution, let's bring all the terms to one side, simplify, and solve for x.

1. Distribute the 1/2 -->
   `(1)/(2) x+6=4x-1`
2. Subtract 4x from both sides -->
   `(1)/(2)x+6-4x=-1`
3. Subtract 6 from both sides -->
   `(1)/(2) x-4x=-7`
4. Combine 1/2x and -4x -->
   `-3.5x=-7`
5. Divide both sides by -3.5 -->
   `x=2`

There only one solution to the equation as shown above and as can be seen in step 4, the x is only to the first degree (
`x=x^1`) which indicates that there is only one solution, 2.

Answer 2

0.5x + 6 = 4x - 1

3.5x = 7

x = 2

Only one solution exists for the given equation.