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

Wierzycaz 2021-10-08 Answered

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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Expert Answer

Cristiano Sears
Answered 2021-10-09 Author has 27241 answers
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 \(\displaystyle{\left(-\infty,\infty\right)}\).
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Relevant Questions

asked 2021-10-11

To solve:
\(\displaystyle{\left|{x}-{2}\right|}{>}{6}\)
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.
We can use the property \(\displaystyle{\left|{x}\right|}{>}{k}\) is equivalent to \(\displaystyle{x}{<}-{k}\ {\quad\text{or}\quad}\ {x}{>}{k}\), where k is a positive number
When solving an absolute value inequality it's necessary to first isolate the absolure value expression on one side of the inequality before solving the inequality.
\(\displaystyle{\left|{a}{x}+{b}\right|}{<}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle=-{c}{<}{a}{x}+{b}{<}{c}\)
\(\displaystyle{\left|{a}{x}+{b}\right|}{>}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle={a}{x}+{b}{<}-{c}\ {\quad\text{or}\quad}\ {a}{x}+{b}{>}{c}\)
We can replace > above with \(\displaystyle\geq\ {\quad\text{and}\quad}{<}{w}{i}{t}{h}\leq\).

asked 2021-10-08

To solve:
\(\displaystyle{\left|{x}+{6}\right|}{>}{0}\)
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.

asked 2021-10-14

To solve:
\(\displaystyle{\left|{3}{x}+{1}\right|}\leq{13}\).
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.
We can use the property \(\displaystyle{\left|{x}\right|}{<}{k}\) is equivalent to \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\), where k is a positive number and we can write a conjuction such as \(\displaystyle{x}\succ{k}\ {\quad\text{and}\quad}\ {x}{<}{k}\) in the compact form.
\(\displaystyle-{k}{<}{x}{<}{k}\).
For example, \(\displaystyle{\left|{x}\right|}{<}{2}\ {\quad\text{and}\quad}\ {\left|{x}\right|}{>}{2}\).
\(\displaystyle{\left|{x}\right|}{<}{2}\), represents the distance between x and 0 that is less than 2.
Whereas \(\displaystyle{\left|{x}\right|}{>}{2}\), represents the distance between x and 0 that is greater than 2.
We can write an absolute value inequality as a compound inequality \(\displaystyle{\left({i}.{e}.\right)}-{2}{<}{x}{<}{2}\).
When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality.
\(\displaystyle{\left|{a}{x}+{b}\right|}{<}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle=-{c}{<}{a}{x}+{b}{<}{c}\)
\(\displaystyle{\left|{a}{x}+{b}\right|}{>}{c}\), where \(\displaystyle{c}{>}{0}\)
\(\displaystyle={a}{x}+{b}{<}-{c}\ {\quad\text{or}\quad}\ {a}{x}+{b}{>}{c}\)
We can replace > above with \(\displaystyle\geq\ {\quad\text{and}\quad}\ {<}\ {w}{i}{t}{h}\ \leq\).

asked 2021-10-23
Solve the following equations and inequalities.
1. \(\displaystyle{\log{{\left({3}{x}-{1}\right)}}}={\log{{\left({4}-{x}\right)}}}\)
2. \(\displaystyle{{\log}_{{{2}}}{x}^{{{3}}}}={{\log}_{{{2}}}{\left({x}\right)}}\)
3. \(\displaystyle{\ln{{8}}}-{x}^{{{2}}}={\ln{{\left({2}-{x}\right)}}}\)
4. \(\displaystyle{{\log}_{{{5}}}{18}}-{x}^{{{2}}}={{\log}_{{{5}}}{\left({6}-{x}\right)}}\)
asked 2021-10-17

To find:
The given statement is true or false.
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.

asked 2021-09-27
Describe the strategy you would use to solve \(\displaystyle{{\log}_{{{6}}}{x}}-{{\log}_{{{6}}}{4}}+{{\log}_{{{6}}}{8}}\).
a. Use the product rule to turn the right side of the equation into a single logarithm. Recognize that the resulting value is equal to x.
b. Express the equation in exponential form, set the exponents equal to each other and solve.
c. Use the fact that the logs have the same base to add the expressions on the right side of the equation together. Express the results in exponential form, set the exponents equal to each other and solve.
d. Use the fact that since both sides of the equations have logarithms with the same base to set the expressions equal to each other and solve.
asked 2021-10-18
For exercise, solve the equations and inequalities. Write the solutions sets to the inequalities in interval notation if possible.
\(\displaystyle{\frac{{{5}}}{{{y}-{4}}}}={\frac{{{3}{y}}}{{{y}+{2}}}}-{\frac{{{2}{y}^{{{2}}}-{14}{y}}}{{{y}^{{{2}}}-{2}{y}-{8}}}}\)

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