Answer :
Certainly! Let's solve the given inequality step by step:
Given the inequality:
[tex]\[ -2 \leq 1 - \frac{2x}{3} \leq \frac{8}{3} \][/tex]
We need to consider this as two separate inequalities and solve each of them independently.
### First Inequality:
[tex]\[ -2 \leq 1 - \frac{2x}{3} \][/tex]
1. Subtract 1 from both sides:
[tex]\[ -2 - 1 \leq - \frac{2x}{3} \][/tex]
[tex]\[ -3 \leq - \frac{2x}{3} \][/tex]
2. Multiply both sides by -3/2 (and remember to reverse the inequality sign when multiplying by a negative number):
[tex]\[ -3 \cdot \left(-\frac{3}{2}\right) \geq \left(-\frac{2x}{3}\right) \cdot \left(-\frac{3}{2}\right) \][/tex]
[tex]\[ \frac{9}{2} \geq x \][/tex]
[tex]\[ x \leq \frac{9}{2} \][/tex]
### Second Inequality:
[tex]\[ 1 - \frac{2x}{3} \leq \frac{8}{3} \][/tex]
1. Subtract 1 from both sides:
[tex]\[ 1 - \frac{2x}{3} - 1 \leq \frac{8}{3} - 1 \][/tex]
[tex]\[ -\frac{2x}{3} \leq \frac{8}{3} - \frac{3}{3} \][/tex]
[tex]\[ -\frac{2x}{3} \leq \frac{5}{3} \][/tex]
2. Multiply both sides by -3/2 (and reverse the inequality sign):
[tex]\[ -\frac{2x}{3} \cdot \left(-\frac{3}{2}\right) \geq \frac{5}{3} \cdot \left(-\frac{3}{2}\right) \][/tex]
[tex]\[ x \geq -1 \cdot -\frac{3}{2} \][/tex]
[tex]\[ x \geq -1 \][/tex]
### Combining the Results:
We now combine the results of the two inequalities:
[tex]\[ -1 \leq x \leq \frac{9}{2} \][/tex]
Finally, we need to find the integer values [tex]\( x \)[/tex] that satisfy the combined range. The only integer values that lie within this range are those for which [tex]\( x \geq -1 \)[/tex] and [tex]\( x \leq \frac{9}{2} \)[/tex].
The integer values [tex]\( x \in \mathbb{W} \)[/tex] (where [tex]\(\mathbb{W}\)[/tex] is the set of whole numbers) that satisfy this condition are 0, 1, 2, 3, and 4:
[tex]\[ \boxed{-1 \text{ and 4 .}} \][/tex]
However, according to the result:
```
([], 9/2, -1.00000000000000)
```
The correct solution indicates there are no whole numbers [tex]\( x \)[/tex] that satisfy the inequality, indicating likely a mistake in stating the problem condition properly. There are no existing solutions in mathematical whole numbers satisfying these conditions.
So, the result might differ in practice, while our theoretical range seems correct.
Given the inequality:
[tex]\[ -2 \leq 1 - \frac{2x}{3} \leq \frac{8}{3} \][/tex]
We need to consider this as two separate inequalities and solve each of them independently.
### First Inequality:
[tex]\[ -2 \leq 1 - \frac{2x}{3} \][/tex]
1. Subtract 1 from both sides:
[tex]\[ -2 - 1 \leq - \frac{2x}{3} \][/tex]
[tex]\[ -3 \leq - \frac{2x}{3} \][/tex]
2. Multiply both sides by -3/2 (and remember to reverse the inequality sign when multiplying by a negative number):
[tex]\[ -3 \cdot \left(-\frac{3}{2}\right) \geq \left(-\frac{2x}{3}\right) \cdot \left(-\frac{3}{2}\right) \][/tex]
[tex]\[ \frac{9}{2} \geq x \][/tex]
[tex]\[ x \leq \frac{9}{2} \][/tex]
### Second Inequality:
[tex]\[ 1 - \frac{2x}{3} \leq \frac{8}{3} \][/tex]
1. Subtract 1 from both sides:
[tex]\[ 1 - \frac{2x}{3} - 1 \leq \frac{8}{3} - 1 \][/tex]
[tex]\[ -\frac{2x}{3} \leq \frac{8}{3} - \frac{3}{3} \][/tex]
[tex]\[ -\frac{2x}{3} \leq \frac{5}{3} \][/tex]
2. Multiply both sides by -3/2 (and reverse the inequality sign):
[tex]\[ -\frac{2x}{3} \cdot \left(-\frac{3}{2}\right) \geq \frac{5}{3} \cdot \left(-\frac{3}{2}\right) \][/tex]
[tex]\[ x \geq -1 \cdot -\frac{3}{2} \][/tex]
[tex]\[ x \geq -1 \][/tex]
### Combining the Results:
We now combine the results of the two inequalities:
[tex]\[ -1 \leq x \leq \frac{9}{2} \][/tex]
Finally, we need to find the integer values [tex]\( x \)[/tex] that satisfy the combined range. The only integer values that lie within this range are those for which [tex]\( x \geq -1 \)[/tex] and [tex]\( x \leq \frac{9}{2} \)[/tex].
The integer values [tex]\( x \in \mathbb{W} \)[/tex] (where [tex]\(\mathbb{W}\)[/tex] is the set of whole numbers) that satisfy this condition are 0, 1, 2, 3, and 4:
[tex]\[ \boxed{-1 \text{ and 4 .}} \][/tex]
However, according to the result:
```
([], 9/2, -1.00000000000000)
```
The correct solution indicates there are no whole numbers [tex]\( x \)[/tex] that satisfy the inequality, indicating likely a mistake in stating the problem condition properly. There are no existing solutions in mathematical whole numbers satisfying these conditions.
So, the result might differ in practice, while our theoretical range seems correct.
