Let's analyze each equation one by one to understand how the value of [tex]\( x \)[/tex] changes as [tex]\( c \)[/tex] decreases, given that [tex]\( c \)[/tex] is positive.
1. Equation: [tex]\( x + c = 0 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x + c = 0 \implies x = -c
\][/tex]
As [tex]\( c \)[/tex] decreases, [tex]\( -c \)[/tex] becomes less negative (or you could say it becomes larger, moving toward zero). Therefore, as [tex]\( c \)[/tex] decreases, the value of [tex]\( x \)[/tex] increases.
2. Equation: [tex]\( -c \cdot x = -c \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
-c \cdot x = -c \implies x = 1
\][/tex]
The value of [tex]\( x \)[/tex] in this equation is always [tex]\( 1 \)[/tex] regardless of the value of [tex]\( c \)[/tex]. Therefore, as [tex]\( c \)[/tex] decreases, the value of [tex]\( x \)[/tex] stays the same.
3. Equation: [tex]\( \frac{x}{c} = 1 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
\frac{x}{c} = 1 \implies x = c
\][/tex]
As [tex]\( c \)[/tex] decreases, [tex]\( x \)[/tex] (which equals [tex]\( c \)[/tex]) also decreases. Therefore, as [tex]\( c \)[/tex] decreases, the value of [tex]\( x \)[/tex] decreases.
To summarize how the value of [tex]\( x \)[/tex] changes as [tex]\( c \)[/tex] decreases for each equation:
| Equation | Value of [tex]\( x \)[/tex] | Change in [tex]\( x \)[/tex] as [tex]\( c \)[/tex] decreases |
|------------------|-------------------|----------------------------------------|
| [tex]\( x + c = 0 \)[/tex] | [tex]\( x = -c \)[/tex] | [tex]\( x \)[/tex] increases |
| [tex]\( -c \cdot x = -c \)[/tex] | [tex]\( x = 1 \)[/tex] | [tex]\( x \)[/tex] stays the same |
| [tex]\( \frac{x}{c} = 1 \)[/tex] | [tex]\( x = c \)[/tex] | [tex]\( x \)[/tex] decreases |
