Question

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

Answer 1

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

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

We'll simplify each of these inequalities step by step.

Step 1: Solve the first inequality

\(-2 ≤ 2x + 13 - 4\) simplifies to:

\[ -2 ≤ 2x + 9 \]

Subtract 9 from both sides:

\[ -11 ≤ 2x \]

Divide by 2:

\[ -\frac{11}{2} ≤ x \]

This can also be written as:

\[ x ≥ -\frac{11}{2} \]

Step 2: Solve the second inequality

The second inequality \(2x + 13 - 4 < 1\) simplifies to:

\[ 2x + 9 < 1 \]

Subtract 9 from both sides:

\[ 2x < -8 \]

Divide by 2:

\[ x < -4 \]

Step 3: Combine the results

Now we combine the results from both inequalities:

\[ -\frac{11}{2} ≤ x < -4 \]

Step 4: Convert to decimal

\(-\frac{11}{2} = -5.5\), so the final inequality is:

\[ -5.5 ≤ x < -4 \]

Step 5: Write the solution in set notation

In set notation, this can be expressed as:

\[ {x : -5.5 ≤ x < -4} \]

Finding the correct answer among the options:

None of your provided options exactly match \(-5.5 ≤ x < -4\). The given options do not seem to reflect the solution we derived. Therefore, it may be necessary to double-check the problem or the options provided.

However, following the steps shown above, the solution to the inequality is definitely:

\[ {x : -\frac{11}{2} ≤ x < -4} \]