Quadratic Equations
Inequalities involving quadratic expressions
Grade 11
Question:
<p>The values of <em>r</em> for which the expression <em>r</em><sup>2</sup> − <em>r</em> − 6 > 0 and <em>r</em><sup>2</sup> − 6<em>r</em> + 5 > 0 simultaneously hold, and also <em>r</em> ≠ 11/5, are:</p>
<p>(a) \(r < -2\)</p>
<p>(b) \(-2 < r < 1\)</p>
<p>(c) \(r > 5\)</p>
<p>(d) \(1 < r < 3\)</p>
Step-by-Step Solution
Key Concept: Solve each quadratic inequality separately, find their intersection, and exclude the given point. The solution is the overlap of two solution sets from quadratic inequalities.
<p><strong>Step 1:</strong> Solve r² − r − 6 > 0</p><p>Factor: (r − 3)(r + 2) > 0</p><p>Critical points: r = −2, r = 3</p><p>Solution: r < −2 or r > 3</p><p></p><p><strong>Step 2:</strong> Solve r² − 6r + 5 > 0</p><p>Factor: (r − 1)(r − 5) > 0</p><p>Critical points: r = 1, r = 5</p><p>Solution: r < 1 or r > 5</p><p></p><p><strong>Step 3:</strong> Find intersection of both conditions</p><p>Condition 1: r < −2 or r > 3</p><p>Condition 2: r < 1 or r > 5</p><p></p><p>Intersection:</p><p>• For r < −2: this satisfies r < 1 ✓ → r < −2</p><p>• For 3 < r < 5: this satisfies r < 1? No. Satisfies r > 5? No. → No solution</p><p>• For r > 5: this satisfies r > 5 ✓ → r > 5</p><p></p><p><strong>Step 4:</strong> Apply restriction r ≠ 11/5</p><p>Since 11/5 = 2.2, which lies in (−2, 3), it's already excluded from r < −2 or r > 5</p><p></p><p>∴ Answer: r < −2 or r > 5 (typically options a and c represent these intervals)</p>
Correct Answer: a, c