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Which set expresses the solution to the compound inequality −2≤2x+13−4<1−2≤2x+13−4<1?(1 point)

Responses

{x:52≤x<7}{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}{x:3≤x<7}

left brace x colon 3 less-than-or-equal-to x less than 7 right brace

{x:−32≤x<1}{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}

1 answer

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

  1. \(-2 \leq 2x + 13 - 4\)
  2. \(2x + 13 - 4 < 1\)

First, simplify the expression:

\[ 2x + 13 - 4 = 2x + 9 \]

Now we have the compound inequality:

\[ -2 \leq 2x + 9 < 1 \]

Now, let's solve both parts.

Solve the left part:

\[ -2 \leq 2x + 9 \] Subtract 9 from both sides:

\[ -2 - 9 \leq 2x \implies -11 \leq 2x \] Divide by 2:

\[ -\frac{11}{2} \leq x \implies x \geq -\frac{11}{2} \]

Solve the right part:

\[ 2x + 9 < 1 \] Subtract 9 from both sides:

\[ 2x < 1 - 9 \implies 2x < -8 \] Divide by 2:

\[ x < -4 \]

Combine the results:

So now we have:

\[ -\frac{11}{2} \leq x < -4 \]

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

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

Thus, the solution can be written in interval notation as:

\[ [-\frac{11}{2}, -4) \]

Now let's check the given options. You specifically asked for an option that matches:

  • \({x: -\frac{11}{2} \leq x < -4}\)

Which response matches?

None of the given responses match our derived solution.

The correct answer would be the set:

\({x: -\frac{11}{2} \leq x < -4 }\).

If you meant to have an option that I didn't see or if there was any confusion, please clarify, and I can help you further!