Question

lesson 7: all, some, or no solutions
cool down: choose your own solution
3x + 8 = 3x+
what value could you write in after 3x that would make the equation true for:

1. no values of x?

9 (any except 8)

1. all values of x?

8

1. just one value of x?

no value

Explanation:

Step1: Analyze no solution case

We start with the equation $3x + 8 = 3x + c$ (where $c$ is the unknown value). Subtract $3x$ from both sides:
$3x - 3x + 8 = 3x - 3x + c$
$8 = c$ would be a true statement, so for no solution, $c
eq 8$ (any number except 8).

Step2: Analyze all solutions case

Using the same equation $3x + 8 = 3x + c$. Subtract $3x$ from both sides:
$3x - 3x + 8 = 3x - 3x + c$
For this to be true for all $x$, we need $8 = c$, so $c=8$.

Step3: Analyze one solution case

Using $3x + 8 = 3x + c$. Subtract $3x$ from both sides:
$8 = c$. This is either always true (all $x$) or always false (no $x$); there is no value of $c$ that leaves a variable term to solve for a single $x$.

Answer:

1. Any real number except 8 (e.g., 9, 0, -2)
2. 8
3. No such value exists