Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A manufacturer's profit P (in thousands of dollars) from selling 'x' units of a certain item is given by P(x) = -x^2 + 10x - 16. For what range of units sold will the manufacturer make a profit (P(x) > 0)?
- Ax < 2 or x > 8
- Bx > 2
- Cx < 8
- D2 < x < 8
Show answer & explanationAnswer & explanation
Correct answer: D. 2 < x < 8
To find when P(x) > 0, first find the roots of P(x) = 0. Set -x^2 + 10x - 16 = 0. Multiply by -1: x^2 - 10x + 16 = 0. Factor this quadratic: (x - 2)(x - 8) = 0. The roots are x = 2 and x = 8. Since the leading coefficient (-1) is negative, the parabola opens downward, meaning the function is above zero between its roots. Thus, profit is made when 2 < x < 8.
Why the other options are wrong
- A. This would be the solution if the parabola opened upwards (P(x) < 0 in between roots).
- B. This only considers one root and assumes profit for all values greater than it.
- C. This only considers one root and assumes profit for all values less than it.
Solving Quadratic Inequalities
Finding the range(s) of x-values for which a quadratic expression is greater than or less than zero.
- First, find the roots (x-intercepts) of the quadratic equation by setting it to zero.
- These roots divide the number line into intervals.
- Test a value from each interval in the original inequality.
- Alternatively, consider the parabola's shape (opens up/down) to determine where it's above/below the x-axis.
Memory trick: Find the roots, check the parabola's smile/frown!