Question

1. complete each equation so that it is true for no va

a. $3x + 6 = 3(x + \underline{\quad\quad})$
b. $x - 2 = -(\underline{\quad\quad} - x)$
c. $\frac{15x - 10}{5} = \underline{\quad\quad} - 2$

Explanation:

Step1: Analyze part a (no solution)

First, expand the right-hand side: $3(x + c) = 3x + 3c$. For the equation $3x + 6 = 3x + 3c$ to have no solution, the constant terms must not be equal, so $3c
eq 6$, meaning $c
eq 2$. We can choose any number except 2, e.g., 1.

Step2: Analyze part b (no solution)

Expand the right-hand side: $-(d - x) = -d + x$. For $x - 2 = x - d$ to have no solution, $-2
eq -d$, so $d
eq 2$. Choose any number except 2, e.g., 3.

Step3: Analyze part c (no solution)

Simplify the left-hand side: $\frac{15x - 10}{5} = 3x - 2$. For $3x - 2 = e - 2$ to have no solution, $3x
eq e$ for all $x$, so $e$ cannot be $3x$. Choose a constant not equal to $3x$, e.g., $2x$.

Answer:

a. $\boldsymbol{1}$ (any number except 2 is valid)
b. $\boldsymbol{3}$ (any number except 2 is valid)
c. $\boldsymbol{2x}$ (any expression not equal to $3x$ is valid)