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Describe the mistake this student made when solving the compound inequality:



$-4 \leq-10-2 x<6$



$\begin{array}{l}

-4 \leq-10-2 x<6 \\

+10+10

\end{array}$



$\begin{array}{l}

\underset{-2}{6} \leq-2 x<16 \\

\div-2<-2 \\

4 \leq x<-8

\end{array}$


AThey divided by a negative number and didn't flip the symbols, and 6 divided by -2 equals -3, not 4.
BThey divided by -2 from all parts and should have divided by 2, and 6 divided by 2 equals 3, not 4.
CThere is no mistake, the student is correct.
DThey added 10 to all parts and should have subtracted.

Verified Answer :

  • The student correctly adds 10 to all parts of the inequality \(-4 \leq -10 - 2x < 6\), resulting in \(6 \leq -2x < 16\).
  • Dividing by \(-2\) requires reversing the direction of the inequality symbols, which the student correctly does to arrive at \(-3 \geq x > -8\).
  • Every step of the student's work is correct, so the final conclusion is that the student made no mistake.
  • The correct option is: \(\boxed{\text{There is no mistake, the student is correct.}}\)

Explanation

  1. Evaluating the Inequality Solution

The goal is to analyze the student's solution to the compound inequality \(-4 \leq -10 - 2x < 6\).

First, we add 10 to all parts of the inequality to isolate the term with \(x\):
\[ -4 + 10 \leq -10 - 2x + 10 < 6 + 10 \]
This simplifies to:
\[ 6 \leq -2x < 16 \]
This step performed by the student is correct.

Next, we divide all parts of the inequality by \(-2\). When dividing or multiplying an inequality by a negative number, the inequality signs must be reversed.
Performing the division:
\[ \frac{6}{-2} \geq \frac{-2x}{-2} > \frac{16}{-2} \]
\[ -3 \geq x > -8 \]
Comparing this result to the student's work:
The student wrote:
\[ -3 \geq x > -8 \]
The student performed the correct operations, applied the correct sign flipping rule, and calculated the quotients correctly. Therefore, the student's solution is correct.

  1. Conclusion

By comparing the student's work to the step-by-step derivation, we see that every operation is mathematically sound. The student correctly added 10 to all parts, correctly divided by \(-2\), and correctly reversed the inequality signs. Thus, there is no mistake in the student's solution. The correct answer is that there is no mistake.

Examples

Compound inequalities are useful in real-world scenarios where a quantity must stay within a specific range, such as temperature control in a laboratory. For instance, if a chemical reaction must be maintained between a certain temperature range to remain stable, a scientist might model the temperature \(T\) based on an equipment setting \(x\). Solving the inequality ensures that the equipment is operated within the safe temperature bounds, preventing hazardous conditions or experimental failure. Mastering these algebraic techniques provides a foundation for accurately predicting and controlling such physical parameters in engineering and science.

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