Straight Lines
Intercepts
Grade 11
Question:
<p>If mb > 0 and the line y = mx + b is given, determine the possible signs of m and b, and analyze the x-intercept.</p>
<p>(a) m > 0, b > 0; x-intercept is positive</p>
<p>(b) m and b have same sign; x-intercept is negative</p>
<p>(c) m > 0, b < 0; x-intercept is positive</p>
<p>(d) m < 0, b < 0; x-intercept is negative</p>
Step-by-Step Solution
Key Concept: When mb > 0, m and b have the same sign, making the x-intercept -b/m always negative.
<p><strong>Step 1:</strong> Given mb > 0, either (m > 0 and b > 0) or (m < 0 and b < 0)</p><p><strong>Step 2:</strong> x-intercept occurs when y = 0: mx + b = 0 → x = -b/m</p><p><strong>Step 3:</strong> Case 1: m > 0, b > 0 → x = -b/m < 0 (negative)</p><p><strong>Step 4:</strong> Case 2: m < 0, b < 0 → x = -b/m = -(-|b|)/(-|m|) < 0 (negative)</p><p><strong>Step 5:</strong> In both cases, x-intercept cannot be positive.</p><p>∴ Answer is (b)</p>
Correct Answer: B