What happens with a sign?

Question

I have something simple to ask, but it often confuses me. What happens with number-sign when it crosses over the equal sign or less-than and greater-than sign. I have an example. $$1-3x\geq0$$ $$-3x\geq-1$$ $$x\geq-1/?$$ Should there be now $-3$ or just $3$? And should maybe a sign $>$ change into $<$?

Answer 1

If you multiply or divide both sides of an inequality by a negative number then you have to flip the sign around, e.g. $$-3x\ge-1\implies x\le-1/(-3) \implies x\le 1/3$$.

If you are adding or subtracting however, you do not need to flip the inequality sign, and this can often make you less uncertain that you are correct by only adding/subtracting rather than multiplying/dividing. For your example,

$$-3x\geq-1\implies 0\ge3x-1\implies1\ge 3x\implies1/3\ge x$$

Answer 2

Experiment with different values around $\pm1/3$ and conclude:

$$\begin{align} &x=-2/3&\to-3x&=2&\color{green}\ge-1\\ &x=-1/3&\to-3x&=1&\color{green}\ge-1\\ &x=-1/6&\to-3x&=1/2&\color{green}\ge-1\\ &x=1/6&\to-3x&=-1/2&\color{green}\ge-1\\ &x=1/3&\to-3x&=-1&\color{green}\ge-1\\ &x=2/3&\to-3x&=-2&\color{red}\ge-1\\ \end{align}$$

Answer 3

Consider this

You know, $2 \lt 3$ right?

But you also know $-2>-3$ by following the number line, having 0 at the centre, as -2 is closer to 0 than -3.

Let's divide the number line into positive space $\forall x\gt0$ and negative space $\forall x \lt 0$. Just remember, when you move from one space to another the equality sign will alter, become opposite, for the equality to hold true.

In your case,

$1-3x \ge0$

Subtract 1 from both sides $-3x \ge-1$

Now, when you will cancel out the minus sign you are changing/moving from one space to another. So, equality will alter.

$3x \le 1$ $\implies x \le \frac{1}{3}$