Answered

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

$\begin{array}{l}

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

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

+10=+10 \\

6 \leq-2 x<16 \\

\div-2+-2 \quad \div-2 \\

4 \leq x<-8

\end{array}$


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

Verified Answer :

  • The student correctly added \(10\) to all parts, resulting in \(6 \leq -2x < 16\).
  • However, when dividing by \(-2\), the student failed to reverse the inequality signs.
  • The student also made an arithmetic error, incorrectly calculating \(6 / -2 = 4\) instead of \(-3\).
  • The correct solution is \(-8 < x \leq -3\), making the fourth option the correct description of the errors.
  • Final Answer: \(\boxed{\text{They divided by a negative number and didn't flip the symbols. And 6 divided by -2 equals -3, not 4.}}\)

Explanation

  1. Analyzing the Student's Work

To analyze the student's work on the compound inequality \(-4 \leq -10 - 2x < 6\), we evaluate each step carefully. The goal is to solve for \(x\). The first step, adding \(10\) to all parts of the inequality, is performed as follows:
\[-4 + 10 \leq -10 - 2x + 10 < 6 + 10\]
\[6 \leq -2x < 16\]
This calculation is correct. The next step is to isolate \(x\) by dividing all parts by \(-2\). When dividing or multiplying an inequality by a negative number, the inequality symbols must be reversed. Specifically, \(\leq\) becomes \(\geq\) and \(<\) becomes \(>\). Applying this rule:
\[\frac{6}{-2} \geq \frac{-2x}{-2} > \frac{16}{-2}\]
\[-3 \geq x > -8\]
This can be rewritten in the standard form as \(-8 < x \leq -3\).

  1. Identifying the Mistakes

Comparing our correct derivation \(-3 \geq x > -8\) (or \(-8 < x \leq -3\)) with the student's result of \(4 \leq x < -8\), we identify two significant errors:

  1. Arithmetic Error: The student calculated \(6 / -2\) as \(4\), but the correct result is \(-3\).
  2. Inequality Symbol Error: The student failed to reverse the inequality signs when dividing by the negative number \(-2\).

Therefore, the correct choice describing the mistakes is: "They divided by a negative number and didn't flip the symbols. And 6 divided by -2 equals -3, not 4."

Examples

When budgeting, if you determine that your remaining debt \(-2x\) must be between \(-6\) and \(-16\), you are essentially solving a compound inequality to understand the required payment range for each of your three accounts. Properly managing inequality signs ensures you don't accidentally invert your financial constraints, which could lead to incorrect budgeting decisions. Recognizing these common errors helps in accurately calculating bounds for various resources.

Related Questions