How many solutions of equation $|3π|x|+5|= 3$ in $\mathbb{R}$.
My work: There are no solutions to this however I do not know how to show it.
Any help will be appreciated.
How many solutions of equation $|3π|x|+5|= 3$ in $\mathbb{R}$.
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algebra-precalculus
absolute-value
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2What makes you _believe_ that there are no solutions? Write that down and you should have a least a first draft of a proof. – 2017-01-21
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0Try showing that the left side must be bigger or equal to 5. – 2017-01-21
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0or just show that the $|\cdot |$ on the left is superfluous – 2017-01-21
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0The answer ive been given is that there are no solutions without workings out. However I have worked out 4 solutions in the real domain which is why im confused @HenningMakholm – 2017-01-21
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1List the solutions you have found. – 2017-01-21
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0x= -2/3π, 2/3π , -8/3π , 8/3π @lulu – 2017-01-21
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0If $x=-\frac 23 \pi$ then your expression is $|3\pi\times \frac 23 \pi +5|=2\pi^2+5 \neq 3$ and so on. – 2017-01-21
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0@user407151: Those are solutions to $\Bigl|3\pi|x|-5\Bigr|=3$, not to $\Bigl|3\pi|x|+5\Bigr|=3$ – 2017-01-21
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0If you meant $x=-\frac 2{3\pi}$ then your expression is $|3\pi \times \frac 2{3\pi}+5|=2+5=7$. – 2017-01-21
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0No, it doesn't. – 2017-01-21
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0Even after all the well-meaning edits, I still find $|3\pi|x|+5|$ unnatural. Is there a better solution than $|3\pi(|x|)+5|$? – 2017-01-21
4 Answers
2
Observe that $ 3 \pi | x | + 5 $ is always positive. Then, $$ | 3 \pi |x| + 5 | = 3 \pi |x| + 5 $$ The ecuation become as $$ 3 \pi |x| + 5 = 3 \longrightarrow 3 \pi |x|= -2 \longrightarrow |x| = -\frac{2}{3 \pi} $$ If $ x > 0 $ we have $$ |x| = x = -\frac{2}{3 \pi} $$ This is absurd because $ -\frac{2}{3 \pi} < 0 $ and we assume that $ x > 0 $.
If $ x < 0 $ we have $$ |x| = -x = -\frac{2}{3 \pi} \longrightarrow x = \frac{2}{3 \pi} $$ This is also absurd because $ \frac{2}{3 \pi} > 0 $ and we assume $ x < 0 $.
The case $ x = 0 $ gives $ |0| = 0 = -\frac{2}{3 \pi} $ but that is absurd. Then there are no solutions. \
It's possible a more efficient argument. In the second equation we arrive to $$ |x| = -\frac{2}{3 \pi} $$ but $ |x| \geq 0 $ because definition of $ | \cdot | $. Then left side is positive but right side is negative. This is absurd.
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5I think after $|x|=-\frac{3}{3\pi}$, it is enough to say "This is impossible because $|x| \ge 0$ for all $x$." – 2017-01-21
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if you meant $$\left|3\pi|x|+5\right|=5$$ then we have two cases:
$$x\geq 0$$ then our inequality is equivaent to $$3\pi x+5=3$$
if $$x<0$$ then we have to solve $$|-3\pi x+5|=3$$
if $$-3\pi x+5\geq0$$ then we get the solution $$x<0$$
if $$-3\pi x+5<0$$ then we have a contradiction since it must be $$\frac{5}{3\pi}
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I guess you did something like $$ 3\pi|x|+5=3\qquad\text{or}\qquad -(3\pi|x|+5)=3 \tag{*} $$ getting, for the first one, $$ 3\pi|x|=-2 $$ that you transformed into $$ 3\pi x=-2\qquad\text{or}\qquad -3\pi x=-2 $$ getting the “solutions” $x=-\frac{2}{3\pi}$ and $x=\frac{2}{3\pi}$.
Working similarly on the second equation in (*), you got $$ 3\pi|x|=-8 $$ that you transformed similarly as before, getting the “solutions” $x=-\frac{8}{3\pi}$ and $x=\frac{8}{3\pi}$.
I wrote “solutions” in inverted commas because these are not solutions of the original equation. If you try plugging in $-\frac{2}{3\pi}$, you indeed get the false equality $$ \left|3\pi\left|-\frac{2}{3\pi}\right|+5\right|=3 $$ It is false because the left-hand side is $7$ and not $3$. You can easily check also the other three purported “solutions” to see that they really aren't.
What you missed is setting also the conditions under which you get the transformed equations. Instead of (*), you should write $$ \begin{cases} 3\pi|x|+5=3\\[4px] 3\pi|x|+5\ge0 \end{cases} \qquad\text{or}\qquad \begin{cases} -(3\pi|x|+5)=3\\[4px] 3\pi|x|+5<0 \end{cases} \tag{**} $$ Now you see that the second group has no solution, because $|x|\ge0$, so $3\pi|x|+5>0$. In the first group, the inequality is likewise redundant, so you get $$ 3\pi|x|=-2 $$ which has no solution again because $|x|\ge0$.
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$$|x| \geq 0$$
$$3\pi |x| \geq 0$$
$$3 \pi |x|+5 \geq 5>3$$
$$|3\pi|x|+5|>3$$
Hence there are no way to get equality.