To find the greatest possible value of [tex]\( y - x \)[/tex] given the constraints,
[tex]\[
-4 \leqslant x \leqslant 2 \quad \text{and} \quad 7 \leqslant x + y \leqslant 12,
\][/tex]
we need to maximize [tex]\( y \)[/tex] while minimizing [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Determine the value of [tex]\( x \)[/tex]:
- Minimize [tex]\( x \)[/tex] to begin. The smallest value [tex]\( x \)[/tex] can take from the range is [tex]\( x = -4 \)[/tex].
2. Determine [tex]\( y \)[/tex]:
- Given [tex]\( x = -4 \)[/tex], use the inequality [tex]\( 7 \leqslant x + y \leqslant 12 \)[/tex].
- If [tex]\( x = -4 \)[/tex]:
[tex]\[
7 \leqslant -4 + y \leqslant 12
\][/tex]
[tex]\[
7 + 4 \leqslant y \leqslant 12 + 4
\][/tex]
[tex]\[
11 \leqslant y \leqslant 16
\][/tex]
- The maximum value [tex]\( y \)[/tex] can take within this new range is [tex]\( y = 16 \)[/tex].
3. Calculate [tex]\( y - x \)[/tex]:
- With [tex]\( x = -4 \)[/tex] and [tex]\( y = 16 \)[/tex],
[tex]\[
y - x = 16 - (-4) = 16 + 4 = 20
\][/tex]
### Conclusion:
The greatest possible value of [tex]\( y - x \)[/tex] is [tex]\( 20 \)[/tex].
Thus, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that achieve this result are [tex]\( x = -4 \)[/tex] and [tex]\( y = 16 \)[/tex].
