Chapter 4.6, Problem 55AYU

In Problems 49-60, solve each inequality algebraically.

$\text{3}\left(x^2\text{ - 2}\right)\text{ < 2}{\left(x\text{ - 1}\right)}^2\text{ + }{x}^2$

To determine

To solve: The inequality algebraically.

Answer to Problem 55AYU

$\left\{x|-\infty <x<2\right\}$

Explanation of Solution

Given:

$3\left(x^2-2\right)<2{\left(x-1\right)}^2+x^2$

Calculation:

$3\left(x^2-2\right)<2{\left(x-1\right)}^2+x^2$

$3x^2-6<2\left(x^2-2x+1\right)+x^2$

$3x^2-6<3x^2-4x+2$

$4x<8$

$x<2$

$x-2<0$

Since we want to know where $f\left(x\right)$ is negative, conclude that $f\left(x\right)<0$. The solution set is $\left\{x|-\infty <x<2\right\}$ or, using interval notation $\left(-\infty ,2\right)$.