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Chapter 3: Problem 14

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 6 x^{2}+x>1 $$

Short Answer

Expert verified
The solution is \((-\infty, -0.28) \cup (0.61, +\infty)\).

Step by step solution

01

Simplify the Polynomial Inequality

First, reshape the inequality to isolate the zero on one side. We obtain the equivalent inequality \(6x^{2} + x - 1 > 0\)
02

Find the Roots

By setting the quadratic equal to zero, we can find the roots of the equation. This means we solve \(6x^{2} + x -1 = 0\). The roots can be found by using the quadratic formula \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here, \(a = 6\), \(b = 1\), and \(c = -1\). Subtracting these values into the formula gives \(x\approx -0.28\) and \(x\approx 0.61\).
03

Test the Intervals & Graph

The roots divide the number line into three regions: \(-\infty,-0.28\), \(-0.28,0.61\), and \(0.61,+\infty\). To find which intervals satisfy the inequality, select a test point from each interval and substitute it back into the inequality. If the inequality is satisfied, that interval is part of the solution. The intervals which satisfy the inequality are \(-\infty,-0.28\) (open) and \(0.61,+\infty\) (open). These intervals can then be graphed on the number line.

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