x lt -22x - 5y gt 31The point \(\mathrm{(-7, y)}\) is a solution to the system of inequalities in the...

Part 1: Brief Solution

Concepts Tested: System of inequalities, coordinate substitution

Primary Process Skills: Translate, Apply Constraints

Essential Steps:

• Substitute \(\mathrm{y = 53}\) into both inequalities to find constraints on x
• From \(\mathrm{y \gt 14}\): \(\mathrm{53 \gt 14}\) ✓ (automatically satisfied)
• From \(\mathrm{4x + y \lt 18}\): \(\mathrm{4x + 53 \lt 18}\) → \(\mathrm{x \lt -8.75}\)
• Only choice A (-9) satisfies \(\mathrm{x \lt -8.75}\)

Answer: A. -9

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Incomplete Constraint Analysis - Phase: Devising Approach → Choice C (5) or D (9)
   • Process skill failure: Apply Constraints
   • Students focus only on \(\mathrm{y \gt 14}\) being satisfied and ignore the second inequality constraint.
2. Arithmetic Error in Inequality Solving - Phase: Executing Approach → Choice B (-5)
   • Computational error: Sign manipulation
   • Students make errors when computing \(\mathrm{18 - 53 = -35}\) or incorrectly solve \(\mathrm{x \lt -8.75}\).
3. Misunderstanding Solution Requirements - Phase: Devising Approach → Various
   • Process skill failure: Translate
   • Students don't recognize that "(x, 53) is a solution" means both x and \(\mathrm{y = 53}\) must satisfy both inequalities simultaneously.

Part 3: Detailed Solution

Understanding the Problem Setup

When we say "the point (x, 53) is a solution to the system of inequalities," this means that when we substitute the coordinates of this point into both inequalities, both must be true simultaneously.

Process Skill: TRANSLATE - We need to convert the statement "point (x, 53) is a solution" into the mathematical requirement that both x and \(\mathrm{y = 53}\) must satisfy both inequalities.

Step 1: Analyze the First Inequality

Starting with \(\mathrm{y \gt 14}\):

Substitute \(\mathrm{y = 53}\):

\(\mathrm{53 \gt 14}\)

This is clearly true regardless of what value x takes. So the first inequality doesn't restrict our choice of x at all.

Step 2: Analyze the Second Inequality

Now for \(\mathrm{4x + y \lt 18}\):

Substitute \(\mathrm{y = 53}\):

\(\mathrm{4x + 53 \lt 18}\)

Process Skill: APPLY CONSTRAINTS - We must work within the boundary established by this inequality to find the valid range for x.

Solving for x:

\(\mathrm{4x \lt 18 - 53}\)

\(\mathrm{4x \lt -35}\)

\(\mathrm{x \lt -35/4}\)

\(\mathrm{x \lt -8.75}\)

Step 3: Check Answer Choices

Process Skill: INFER - Now we need to determine which of the given choices satisfies our constraint \(\mathrm{x \lt -8.75}\).

Examining each choice:

• A. -9: Is \(\mathrm{-9 \lt -8.75}\)? Yes! ✓
• B. -5: Is \(\mathrm{-5 \lt -8.75}\)? No, \(\mathrm{-5 \gt -8.75}\) ✗
• C. 5: Is \(\mathrm{5 \lt -8.75}\)? No, \(\mathrm{5 \gt -8.75}\) ✗
• D. 9: Is \(\mathrm{9 \lt -8.75}\)? No, \(\mathrm{9 \gt -8.75}\) ✗

Verification

Let's verify that \(\mathrm{x = -9}\) works by substituting back:

• First inequality: \(\mathrm{y \gt 14}\) → \(\mathrm{53 \gt 14}\) ✓
• Second inequality: \(\mathrm{4x + y \lt 18}\) → \(\mathrm{4(-9) + 53 = -36 + 53 = 17 \lt 18}\) ✓

Both inequalities are satisfied, confirming our answer.

The answer is A. -9

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

Incomplete Constraint Analysis (Process Skill: Apply Constraints)

Many students recognize that \(\mathrm{y = 53}\) satisfies \(\mathrm{y \gt 14}\), but then assume this means any x-value will work. They fail to recognize that the second inequality \(\mathrm{4x + y \lt 18}\) creates a significant restriction on x. This leads them to pick positive values like 5 or 9, not realizing these violate the constraint \(\mathrm{x \lt -8.75}\).

Misunderstanding Solution Requirements (Process Skill: Translate)

Some students don't fully grasp what "(x, 53) is a solution to the system" means. They might try to solve for both x and y simultaneously, or they might not understand that they need to substitute \(\mathrm{y = 53}\) into both inequalities. This conceptual gap prevents them from setting up the problem correctly from the start.

Errors while executing the approach:

Arithmetic Error in Inequality Solving (Computational Error)

When solving \(\mathrm{4x + 53 \lt 18}\), students commonly make arithmetic mistakes:

• Computing \(\mathrm{18 - 53}\) incorrectly (getting -25 instead of -35)
• Dividing -35 by 4 incorrectly (getting -8.25 instead of -8.75)
• Making sign errors when isolating x

These computational errors lead to incorrect boundary values, causing students to select choice B (-5) which seems close to the correct boundary.

Inequality Direction Confusion (Computational Error)

Some students correctly compute \(\mathrm{x \lt -8.75}\) but then mistakenly think this means x must be greater than -8.75, leading them to eliminate choice A and select one of the positive options.

Errors while selecting the answer:

Boundary Value Misinterpretation (Process Skill: Apply Constraints)

Even when students correctly find \(\mathrm{x \lt -8.75}\), they might not carefully check which answer choices actually satisfy this constraint. They might assume that -5 is "close enough" to -8.75 without recognizing that \(\mathrm{-5 \gt -8.75}\) violates the inequality.

Verification Skip (Process Skill: Infer)

Students who arrive at the correct constraint might select choice A without verifying their work, missing the opportunity to catch any computational errors they made along the way. While this doesn't lead to an incorrect answer in this case, it's a risky approach that could cause problems in other questions.