Let's break down and solve the inequalities step-by-step to find the constraints on [tex]\( w \)[/tex] (the width of the rectangular pool). We are given the following system of inequalities:
1. [tex]\( w \leq 10 - 1 \)[/tex]
2. [tex]\( 21 + 2w \geq 62 \)[/tex]
First, let's solve each inequality individually.
Step 1: Solve [tex]\( w \leq 10 - 1 \)[/tex]
[tex]\[ w \leq 9 \][/tex]
So, the first constraint is [tex]\( w \leq 9 \)[/tex].
Step 2: Solve [tex]\( 21 + 2w \geq 62 \)[/tex]
First, isolate the term involving [tex]\( w \)[/tex]:
[tex]\[ 2w \geq 62 - 21 \][/tex]
[tex]\[ 2w \geq 41 \][/tex]
Next, solve for [tex]\( w \)[/tex] by dividing both sides by 2:
[tex]\[ w \geq \frac{41}{2} \][/tex]
[tex]\[ w \geq 20.5 \][/tex]
So, the second constraint is [tex]\( w \geq 20.5 \)[/tex].
Combining the Results:
We need [tex]\( w \leq 9 \)[/tex] and [tex]\( w \geq 20.5 \)[/tex] to hold simultaneously. However, examining these two inequalities shows that there are no values of [tex]\( w \)[/tex] that can simultaneously satisfy [tex]\( w \leq 9 \)[/tex] and [tex]\( w \geq 20.5 \)[/tex]. Hence, there is no range of [tex]\( w \)[/tex] values that satisfy both conditions together.
This inconsistency means that within the context of these constraints, there are no possible values for [tex]\( w \)[/tex] that can describe the width of the rectangular pool.
