Question

| 3/x - 2 | <=5Step by step

Answer 1

So we need to solve the following inequality:

`\lvert(3)/(x)-2\rvert\leq5`

The first thing to do here is getting rid of the absolute value. Remember what happen when you remove an absolute value in an inequality:

`\begin{gathered} \lvert a\rvert\leq b \\ a\leq b\text{ and }a\ge-b \end{gathered}`

So we'll have two inequalities. The one multiplied by a negative sign must have the other inequality symbol. Taking this into account let's get back to our problem:

`\begin{gathered} \lvert(3)/(x)-2\rvert\leq5 \\ (3)/(x)-2\leq5\text{ and }(3)/(x)-2\ge-5 \end{gathered}`

Let's solve each separately. We take the first one and we add 2 to both sides:

`\begin{gathered} (3)/(x)-2\leq5 \\ (3)/(x)-2+2\leq5+2 \\ (3)/(x)\leq7 \end{gathered}`

Then we multiply x to both sides but first we need to note that if x is negative then we have to change the inequality symbol. So we have two cases for this inequality, x>0 and x<0.

If x>0:

`\begin{gathered} (3)/(x)\leq7 \\ (3)/(x)\cdot x\leq7x \\ 3\leq7x \end{gathered}`

And we divide by 7:

`\begin{gathered} (3)/(7)\leq(7x)/(7) \\ (3)/(7)\leq x \end{gathered}`

If x<0 we have:

`\begin{gathered} (3)/(x)\leq7 \\ (3)/(x)\cdot x\ge7x \\ 3\ge7x \end{gathered}`

We divide both sides by 7 and we get:

`(3)/(7)\ge x`

Then we solve the other inequality. We add 2 to both sides:

`\begin{gathered} (3)/(x)-2+2\ge-5+2 \\ (3)/(x)\ge-3 \end{gathered}`

And we can multiply both sides by x. If x>0 we get:

`\begin{gathered} (3)/(x)\cdot x\ge-3x \\ 3\ge-3x \end{gathered}`

Then we divide by -3. Remember that multiplying or dividing by a negative number means that we have to change the inequality symbol:

`\begin{gathered} (3)/(-3)\leq(-3x)/(-3) \\ -1\leq x \end{gathered}`

If x<0 we have:

`\begin{gathered} (3)/(x)\cdot x\leq-3x \\ 3\leq-3x \end{gathered}`

Then we divide by -3. Remember that multiplying or dividing by a negative number means that we have to change the inequality symbol:

`\begin{gathered} (3)/(-3)\ge-(3x)/(-3) \\ -1\ge x \end{gathered}`

Now let's take everything we have. If x<0 we found that:

`\begin{gathered} (3)/(7)\ge x \\ -1\ge x \end{gathered}`

Since 3/7>-1 we just need to use the second inequlity:

`-1\ge x`

For x>0 we got:

`\begin{gathered} (3)/(7)\leq x \\ -1\leq x \end{gathered}`

Since 3/7>-1 we just need to use the first inequality:

`(3)/(7)\leq x`

Then:

`x\leq-1\text{ and }(3)/(7)\leq x`

And that's the answer.