Solve the inequality:
$-2 x^2 + 24 x - 70 > 0$
Solution Set: ____
The problem asks us to solve the quadratic inequality \(-2x^2 + 24x - 70 > 0\). We are dealing with a downward-opening parabola, which means the values of the quadratic expression are positive between its two roots. Our first task is to simplify the inequality and identify those roots.
To simplify the inequality, we can divide the entire expression by \(-2\). Recall that when dividing an inequality by a negative number, the direction of the inequality sign must be reversed. This gives us:
\[ \frac{-2x^2 + 24x - 70}{-2} < \frac{0}{-2} \]
\[ x^2 - 12x + 35 < 0 \]
We now look for the roots of the quadratic equation \(x^2 - 12x + 35 = 0\). We can factor this expression by looking for two numbers that multiply to \(35\) and add up to \(-12\). These numbers are \(-5\) and \(-7\). Thus, we can factor the quadratic as:
\[ (x - 5)(x - 7) < 0 \]
The roots are \(x = 5\) and \(x = 7\).
Since the quadratic expression \(x^2 - 12x + 35\) represents an upward-opening parabola, it is negative (less than zero) strictly between its two roots, \(5\) and \(7\). Therefore, the inequality holds for all real numbers \(x\) such that \(5 < x < 7\). In interval notation, the solution set is \((5, 7)\).
Quadratic inequalities like this one are frequently encountered in engineering and economics when determining the range of values for which a specific outcome is positive or profitable. For example, if a company's profit \(P\) from producing \(x\) units is modeled by the function \(P(x) = -2x^2 + 24x - 70\), solving this inequality helps identify the production range in thousands of units where the company remains profitable, which in this case is between \(5,000\) and \(7,000\) units.