Question

consider the system of inequalities and its graph.

\\(y \le -0.75x\\)
\\(y \le 3x - 2\\)

in which section of the graph does the actual solution to the system lie?

Explanation:

Analyze the first inequality

Using the Systems of Linear Inequalities knowledge point

$$ y \le -0.75x $$

The boundary line is \(y = -0.75x\). Since the inequality is \(\le\), the solution region lies on or below this boundary line.

Analyze the second inequality

Using the Systems of Linear Inequalities knowledge point

$$ y \le 3x - 2 $$

The boundary line is \(y = 3x - 2\). Since the inequality is \(\le\), the solution region lies on or below this boundary line.

Test a point in each region

Using the Constraint Verification knowledge point

• Region 1: Test point \((-6, -4)\)
• \( -4 \le -0.75(-6) \implies -4 \le 4.5 \) (True)
• \( -4 \le 3(-6) - 2 \implies -4 \le -20 \) (False)
• Region 2: Test point \((-2, 4)\)
• \( 4 \le -0.75(-2) \implies 4 \le 1.5 \) (False)
• Region 3: Test point \((6, 2)\)
• \( 2 \le -0.75(6) \implies 2 \le -4.5 \) (False)
• Region 4: Test point \((4, -4)\)
• \( -4 \le -0.75(4) \implies -4 \le -3 \) (True)
• \( -4 \le 3(4) - 2 \implies -4 \le 10 \) (True)

Identify the solution section

Using the Constraint Verification knowledge point
The only region satisfying both inequalities is Section 4.

Answer:

• 2
• 4 (Correct answer)
• 1
• 3