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Chapter 2: Problem 67

Solve the inequality. (Round your answers to two decimal places.) $$0.4 x^{2}+5.26<10.2$$

Short Answer

Expert verified
The solution to the inequality is \(x\in(-3.51, 3.51)\). Please note that these values are rounded to two decimal places.

Step by step solution

01

- Simplify the Inequality to Standard Form

Subtract 10.2 from both sides of the inequality to set it equal to zero and bring it into a standard form - \(0.4 x^{2}+5.26-10.2<0$ which simplifies to \(0.4 x^{2}-4.94<0\).
02

- Solve the Quadratic Equation

This step requires solving the corresponding quadratic equation \(0.4 x^{2}-4.94=0\) for x. Divide the whole equation by 0.4 to simplify the equations to \(x^{2}-12.35=0\). Use the quadratic formula to find the roots x1 and x2 : \(x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). This gives us \(x_{1,2}=\pm\sqrt{12.35}\).
03

- Determine the Solution Interval

With the roots found, the solutions of the inequality split into three intervals: \(-\infty\sqrt{12.35}\). Test a number from each interval in the original inequality to determine which intervals are solutions. The solution turns out to be the interval \(-\sqrt{12.35}<x<\sqrt{12.35}\)

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