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Which set expresses the solution to the compound inequality −2≤2x+13−4<1?(1 point)Responses{x:−32≤x<1}left brace x colon Start Fraction negative 3 over 2 End Fraction less-than-or-equal-to x less than 1 right brace{x:52≤x<1}left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 1 right brace{x:52≤x<7}left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 7 right brace{x:3≤x<7}

1 answer

To solve the compound inequality \(-2 \leq 2x + 13 - 4 < 1\), we can break it down into two parts:

  1. Solve the left part: \(-2 \leq 2x + 13 - 4\)
  2. Solve the right part: \(2x + 13 - 4 < 1\)

Let's solve each part step by step.

Step 1: Solve the left part

\[ -2 \leq 2x + 13 - 4 \] Simplify the expression on the right: \[ -2 \leq 2x + 9 \] Now, isolate \(2x\): \[ -2 - 9 \leq 2x \quad \Rightarrow \quad -11 \leq 2x \] Dividing both sides by 2 gives: \[ -\frac{11}{2} \leq x \quad \Rightarrow \quad x \geq -\frac{11}{2} \]

Step 2: Solve the right part

\[ 2x + 13 - 4 < 1 \] Simplifying the right-hand side: \[ 2x + 9 < 1 \] Isolate \(2x\): \[ 2x < 1 - 9 \quad \Rightarrow \quad 2x < -8 \] Dividing both sides by 2 gives: \[ x < -4 \]

Combined result

Now we have: \[ -\frac{11}{2} \leq x < -4 \]

Convert \(-\frac{11}{2}\) to a decimal:

\(-\frac{11}{2} = -5.5\).

The solution set:

We need to express the solution set in correct notation.

So, the solution combines as: \[ -\frac{11}{2} \leq x < -4 \text{ or } -5.5 \leq x < -4. \]

Check options available:

The set should be in proper setup. The intervals given in the responses are:

  1. {x: -32 ≤ x < 1}
  2. {x: 52 ≤ x < 1}
  3. {x: 52 ≤ x < 7}
  4. {x: 3 ≤ x < 7}

None of the above options appear to be correct, as none match our derived solution.

If the problem intended to say that the options were mistyped or misrepresented, you might need to correct the error or re-evaluate the proper options reflecting the actual derived solutions. As it stands, based on the solution derived, none of the provided options show the right interpretation for the compound inequality solution.