Question

amelia, luis, shauna, and clarence used different approaches to solve the inequality

(7.2b + 6.5 > 4.8b - 8.1).

• amelia started by subtracting (7.2b) from both sides to get (6.5 > -2.4b - 8.1).
• luis started by subtracting (4.8b) from both sides to get (2.4b + 6.5 < -8.1).
• shauna started by subtracting (6.5) from both sides to get (7.2b > 4.8b - 14.6).
• clarence started by adding (8.1) to both sides to get (7.2b + 14.6 > 4.8b).

which students first step was incorrect, and why?

• amelias, because the variable term must be isolated on the left side
• shaunas, because she did not apply the subtraction property of equality properly
• clarences, because the terms he added together were not like terms
• luiss, because he flipped the inequality sign when he subtracted

Explanation:

Analyze the original inequality

Using the Solving Linear Inequalities knowledge point
The original inequality is:

$$7.2b + 6.5 > 4.8b - 8.1$$

Evaluate Amelia's first step

Using the Solving Linear Inequalities knowledge point
Amelia subtracts \(7.2b\) from both sides:

$$(7.2b - 7.2b) + 6.5 > (4.8b - 7.2b) - 8.1$$

$$6.5 > -2.4b - 8.1$$

This step is mathematically correct.

Evaluate Luis's first step

Using the Solving Linear Inequalities knowledge point
Luis subtracts \(4.8b\) from both sides:

$$(7.2b - 4.8b) + 6.5 > (4.8b - 4.8b) - 8.1$$

$$2.4b + 6.5 > -8.1$$

However, Luis wrote:

$$2.4b + 6.5 < -8.1$$

He flipped the inequality sign. Subtracting a term does not change the inequality direction. Thus, Luis's step is incorrect.

Evaluate Shauna's first step

Using the Solving Linear Inequalities knowledge point
Shauna subtracts \(6.5\) from both sides:

$$7.2b + 6.5 - 6.5 > 4.8b - 8.1 - 6.5$$

$$7.2b > 4.8b - 14.6$$

This step is mathematically correct.

Evaluate Clarence's first step

Using the Solving Linear Inequalities knowledge point
Clarence adds \(8.1\) to both sides:

$$7.2b + 6.5 + 8.1 > 4.8b - 8.1 + 8.1$$

$$7.2b + 14.6 > 4.8b$$

This step is mathematically correct.

Answer:

• Amelia's, because the variable term must be isolated on the left side
• Shauna's, because she did not apply the subtraction property of equality properly
• Clarence's, because the terms he added together were not like terms
• Luis's, because he flipped the inequality sign when he subtracted (Correct answer)