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Chapter 2: Problem 68

Solve the inequality. (Round your answers to two decimal places.) $$-1.3 x^{2}+3.78>2.12$$

Short Answer

Expert verified
The solution to the inequality \(-1.3x^2 + 3.78 > 2.12\) is \(x 鈭� (-1.18, 1.18)\)

Step by step solution

01

Simplify the inequality

Subtract 2.12 from both sides of the inequality to isolate the quadratic expression on one side. So, the inequality becomes: \(-1.3x^2 + 3.78 - 2.12 > 0\), which simplifies to \(-1.3x^2 + 1.66 > 0\)
02

Solve the associated equation

Solving the equation \(-1.3x^2 + 1.66 = 0\) gives the critical points. Rearranging the equation gives \(x^2 = \frac{1.66}{1.3}\). Taking the square root of both sides gives \(x = \pm\sqrt{\frac{1.66}{1.3}}\)
03

Find the interval

Subtract and add \(\sqrt{\frac{1.66}{1.3}}\) in the inequality \(-1.3x^2 + 1.66 > 0\) which gives you \(x 鈭� (-\sqrt{\frac{1.66}{1.3}}, \sqrt{\frac{1.66}{1.3}})\)
04

Round off to two decimal places

The above interval rounds off to \(x 鈭� (-1.18, 1.18)\)

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