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Chapter 2: Problem 69

Solve the inequality. (Round your answers to two decimal places.) $$-0.5 x^{2}+12.5 x+1.6>0$$

Short Answer

Expert verified
\(0.12 < x < 26.78\)

Step by step solution

01

Rewrite the inequality

Rewrite the inequality in standard form: \(-0.5 x^{2}+12.5 x+1.6 > 0\) can be rewritten as \(0.5 x^{2} - 12.5 x - 1.6 < 0\)
02

Find the roots

Next, to find the roots of the inequality, use the quadratic formula \(x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substitute \(a=0.5\), \(b=-12.5\), and \(c=-1.6\) into the formula, yielding the roots \(x=0.12\) and \(x=26.78\)
03

Find the intervals

Now, treat the roots found as potential boundaries where the inequality might change sign. These form three intervals: \(-\infty < x < 0.12\)(interval 1), \(0.12 < x < 26.78\)(interval 2), \(26.78 < x < \infty\)(interval 3). Check the sign of the inequality by substituting a value from each interval into the inequality.
04

Check the sign

Choose \(x=0\) for interval 1, \(x=1\) for interval 2, and \(x=30\) for interval 3, and substitute these values into the inequality. We find that for interval 1 the inequality is positive, for interval 2 the inequality is negative, and for interval 3 the inequality is positive.
05

Write the final solution

The solution to the inequality is where the quadratic is less than 0, i.e. from the negative intervals. Therefore, the solution to the inequality is \(0.12 < x < 26.78\).

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