How to Treat this Absolute Inequality

[Mathdabbler]

I’ve tried to solve this inequality [the correct answer is given as (-1)] but it seems I am not treating the 4 in the equation correctly. Do I assume the 4 is multiplied by the part equation in the absolute brackets? Or is it multiplied by the 3/5 term only?

Any help to point me in the right direction will be appreciated

 

[MarkFL]

The 4 is a coefficient for the entire expression within the absolute value. I would factor as follows:

[MATH]2\left|n-\frac{6}{5}\right|>1[/MATH]
And then:

[MATH]\left|n-\frac{6}{5}\right|>\frac{1}{2}[/MATH]
This tells me that \(n\) is the set of all numbers on the real number line whose distance from [MATH]\frac{6}{5}[/MATH] is greater than [MATH]\frac{1}{2}[/MATH] or:

[MATH]\left(-\infty,\frac{6}{5}-\frac{1}{2}\right)\,\cup\,\left(\frac{6}{5}+\frac{1}{2},\infty\right)[/MATH]
[MATH]\left(-\infty,\frac{7}{10}\right)\,\cup\,\left(\frac{17}{10},\infty\right)[/MATH]

 

[Dr.Peterson]

Alternatively, you could divide both sides by 4 to isolate the absolute value as it stands, and then use whatever your standard method is for solving such inequalities.

But how could the solution be a single number, "(-1)"? Are you sure you looked at the right answer?

 

[Mathdabbler]

Ok - that’s made it clear ie treat the 4 as the coefficient for the entire expression within the absolute bars.

Apologies I did put the wrong answer up of -1

 

[pka]

I’ve tried to solve this inequality [the correct answer is given as (-1)] but it seems I am not treating the 4 in the equation correctly. Do I assume the 4 is multiplied by the part equation in the absolute brackets? Or is it multiplied by the 3/5 term only?

You may find this a confusing after-thought, if so please ignore.
Two properties of absolute value: since the distance from \(\displaystyle x\text{ to }y\) is \(\displaystyle |x-y|\) so that \(\displaystyle |x-y|=|y-x|\). And \(\displaystyle |x\cdot y|=|x|\cdot |y|\).
So I would rewrite \(\displaystyle 4\left|\frac{3}{5}-\frac{n}{2}\right|=4\left|\frac{n}{2}-\frac{3}{5}\right|\). then
\(\displaystyle 4\left|\frac{n}{2}-\frac{3}{5}\right|>1\text{ divide by }4\\\left|\dfrac{n}{2}-\dfrac{3}{5}\right|>0.25\text{ now, multiply through by }10\\\left|5n-6\right|>2.5\)
At this point it is an inequality is in standard(usual) form.