A curve is described by the equation \(y = 2x^2 - 6x + 5\). Determine the range of \(x\) values for which \(y > 13\).
Solution
Start with the inequality:
\(2x^2 - 6x + 5 > 13\)
Simplify the inequality:
\(2x^2 - 6x + 5 - 13 > 0\)
\(2x^2 - 6x - 8 > 0\)
Factor the quadratic:
\(2(x^2 - 3x - 4) > 0\)
\(2(x - 4)(x + 1) > 0\)
Find the critical points by setting the factors to zero:
\(x - 4 = 0 \Rightarrow x = 4\)
\(x + 1 = 0 \Rightarrow x = -1\)
Test intervals around the critical points:
- For \(x < -1\), choose \(x = -2\): \(2(-2 - 4)(-2 + 1) > 0 \Rightarrow 2(-6)(-1) > 0 \Rightarrow 12 > 0\)
- For \(-1 < x < 4\), choose \(x = 0\): \(2(0 - 4)(0 + 1) > 0 \Rightarrow 2(-4)(1) > 0 \Rightarrow -8 > 0\) (false)
- For \(x > 4\), choose \(x = 5\): \(2(5 - 4)(5 + 1) > 0 \Rightarrow 2(1)(6) > 0 \Rightarrow 12 > 0\)
Thus, the solution is \(x > 4\) or \(x < -1\).
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