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Which set expresses the solution to the compound inequality $-2\leqslant \frac {2x+1}{3}-4\lt 1$ ? (1 point)

$\{ x:\frac {5}{2}\leqslant x\lt 1\} $
$\{ x:\frac {5}{2}\leqslant x\lt 7\} $
$\{ x:\frac {-3}{2}\leqslant x\lt 1\}$
$\{ x:3\leqslant x\lt 7\} $

1 answer

To solve the compound inequality \( -2 \leq \frac{2x+1}{3} - 4 < 1 \), we will break it down into two parts: the left side \( -2 \leq \frac{2x+1}{3} - 4 \) and the right side \( \frac{2x+1}{3} - 4 < 1 \).

Step 1: Solve the left side \( -2 \leq \frac{2x+1}{3} - 4 \)

  1. First, we isolate \( \frac{2x+1}{3} \): \[ -2 + 4 \leq \frac{2x+1}{3} \] \[ 2 \leq \frac{2x+1}{3} \]

  2. Multiply both sides by 3: \[ 6 \leq 2x + 1 \]

  3. Subtract 1 from both sides: \[ 5 \leq 2x \]

  4. Divide by 2: \[ \frac{5}{2} \leq x \]

Step 2: Solve the right side \( \frac{2x+1}{3} - 4 < 1 \)

  1. Isolate \( \frac{2x+1}{3} \): \[ \frac{2x+1}{3} < 1 + 4 \] \[ \frac{2x+1}{3} < 5 \]

  2. Multiply both sides by 3: \[ 2x + 1 < 15 \]

  3. Subtract 1 from both sides: \[ 2x < 14 \]

  4. Divide by 2: \[ x < 7 \]

Combining the results

Now, we combine the results of both parts: \[ \frac{5}{2} \leq x < 7 \]

Answer as a set

This can be expressed as a set: \[ { x : \frac{5}{2} \leq x < 7 } \]

Final answer

Among the choices provided, the correct one is: \[ { x : \frac{5}{2} \leq x < 7 } \]