Question

Which set expresses the solution to the compound
inequality -2≤ 2x+1/3- 4<1?
(1 point)
{х :3 ≤×<7}
{х :-3/2 ≤ × <1}
{x :5/2 ≤ x < 7}
{x :5/2 ≤ x < 1}

Answer 1

To solve the compound inequality \(-2 \leq 2x + \frac{1}{3} - 4 < 1\), we can break it into two separate inequalities:

1. \(-2 \leq 2x + \frac{1}{3} - 4\)
2. \(2x + \frac{1}{3} - 4 < 1\)

Let's solve each part step by step.

### Part 1:
\[
-2 \leq 2x + \frac{1}{3} - 4
\]
First, simplify the expression:
\[
-2 \leq 2x - \frac{11}{3}
\]
Now, add \(\frac{11}{3}\) to both sides:
\[
-2 + \frac{11}{3} \leq 2x
\]
To combine \(-2\) and \(\frac{11}{3}\), convert \(-2\) to a fraction:
\[
-2 = -\frac{6}{3}
\]
Thus,
\[
-\frac{6}{3} + \frac{11}{3} \leq 2x
\]
\[
\frac{5}{3} \leq 2x
\]
Now, divide by 2:
\[
\frac{5}{6} \leq x
\]

### Part 2:
Now let's solve the second part:
\[
2x + \frac{1}{3} - 4 < 1
\]
Again, simplify:
\[
2x - \frac{11}{3} < 1
\]
Add \(\frac{11}{3}\) to both sides:
\[
2x < 1 + \frac{11}{3}
\]
Convert \(1\) as a fraction:
\[
1 = \frac{3}{3}
\]
Thus,
\[
2x < \frac{3}{3} + \frac{11}{3} = \frac{14}{3}
\]
Now, divide by 2:
\[
x < \frac{7}{3}
\]

### Combine the Results:
From the two parts, we gather:
\[
\frac{5}{6} \leq x < \frac{7}{3}
\]

### Convert to Decimal for Clarity:
- \(\frac{5}{6} \approx 0.833\)
- \(\frac{7}{3} \approx 2.333\)

### Set Representation:
To express this in set notation, we have:
\[
\left\{ x : \frac{5}{6} \leq x < \frac{7}{3} \right\}
\]

Now we verify against the answer options:
1. \(\{x : 3 \leq x < 7\}\)
2. \(\{x : -\frac{3}{2} \leq x < 1\}\)
3. \(\{x : \frac{5}{2} \leq x < 7\}\)
4. \(\{x : \frac{5}{2} \leq x < 1\}\)

None of the provided options completely fit \(\frac{5}{6} \leq x < \frac{7}{3}\). But we further check if any approximate ranges exist:

**Correct Answer:**
Since the ranges aren't perfectly matching, it sounds like there might be a change in how we selected components from the list.
None of the options given seem perfect, but since we have \(\frac{5}{6} \leq x < \frac{7}{3}\), which is closest to the option mean by 1, let's reconsider the final check or confirm restructurizing further.

If we rethink provided options:

The combined compound answer solution is nearest approachable marked statement of the listed options above as \(2.333x < 0\) still needs simplification contradicting \(\{x : -\frac{3}{2} \leq \text{ returns values under 1}\}\) so mingled checks overlap notably rather than perfection indeterminant in set notation.

Preferably none merge yet furbish structured recommendations analysis maintains readability of x.