0

Here the RHS is undefined when $x=0$, so the set of all points on the $y$-axis should be excluded (so a dotted line should be drawn over the $y$-axis).

What about the set of all points on the $x$-axis? Should they be included or excluded?

I do not see why they should be excluded. If we test $(-1,0)$, $0 \geq \frac{1}{-1}$ so $(-1,0)$ should be included.

But in my textbook, they have drawn a dotted line over the $x$-axis, meaning all points on the $x$-axis (such as $(-1,0)$) are excluded, right?

Why is this? Could someone explain? Is this because when we graph $y=\frac{1}{x}$, the $x$-axis is a horizontal asymptote?

Freddie
  • 461
  • So far, I agree: some points (not all) on the $x$-axis should be included in the shaded region. For the sake of clarity, could you please provide a picture of your textbook's diagram? – Theo Bendit Apr 12 '21 at 13:19
  • Points on the positive $x$-axis should be excluded; points on the negative $x$-axis should be included – J. W. Tanner Apr 12 '21 at 13:20

1 Answers1

1

Consider the equation $y\geq \frac{1}{x}$. We will convert it into a form that's easier to work with. Note that when we are multiplying both sides of an inequality with a negative number, we need to change it's sign.We will multiply both sides of the equation you have with $x$ \begin{align*} y\geq \frac{1}{x}&\equiv \begin{cases}xy\geq x\frac{1}{x}&x\geq 0\\xy\leq x\frac{1}{x}&x<0 \end{cases}&\equiv \begin{cases}xy\geq 1&x\geq 0\\xy\leq 1&x<0 \end{cases} \end{align*}

Now you can see that in the positive quadrant, it includes all points "above" and including $xy=1$.

When $x<0$, it includes all points between $xy=1$ and the negative x and y axes, inclusive of all of these three curves.

Rahul Madhavan
  • 2,789
  • 1
  • 11
  • 14