# To solve:|3x-1|>-2General strategy to solve the inequalities tha

To solve:
$$\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}$$
General strategy to solve the inequalities that involve absolute value:
Absolute value inequalities deal with the inequalities $$\displaystyle{\left({<},{>},\leq,{\quad\text{and}\quad}\geq\right)}$$ on the expressions with absolute sign.
General properties for solving inequalities that involve absolute value are
1) $$\displaystyle{\left|{x}\right|}{>}{k}$$ is equivalent to $$\displaystyle{x}{<}-{k}\ {\quad\text{or}\quad}\ {x}{>}{k}$$, where k is positive number.
2) $$\displaystyle{\left|{x}\right|}{<}{k}$$ is equivalent to $$\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}$$, where k is a positive number and we can write a conjunction such as $$\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}$$ in the compact form $$\displaystyle-{k}{<}{x}{<}{k}$$.
If k is a non positive number, we can determine the solution sets by inspection.

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Calculation:
To solve: $$\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}$$
The given equation is of the form of $$\displaystyle{\left|{x}\right|}{>}{k}$$. However, the standard method for solving inequalities that involve absolute value cannot be applied to $$\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}$$ because k is negative.
We will solve the given equation $$\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}$$ by inspection.
$$\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}$$ is satisfied by all real numbers because the absolute value of $$\displaystyle{\left({3}{x}-{1}\right)}$$, regardless of what number is substituted for x, will always be greater than -1.
The solution set is the set of all real numbers, which we can express in interval notation as $$\displaystyle{\left(-\infty,\infty\right)}$$.
Conclusion:
Absolute values are always greater or equal to zero. So $$\displaystyle{\left|{3}{x}-{1}\right|}\succ{2}$$ is true for all x and the interval notation is $$\displaystyle{\left(-\infty,\infty\right)}$$.