# How is using inequalities $\ge$ and $\le$ different for being used to solve equations?

I seriously want to know the difference for finding the solution to an equation and an inequality using $\ge$ and $\le$. I know how to solve inequalities involving variables like $9x+5$ $\ge$ $42$, but how is this different from solving and equation? I just need your help for this.

It is possible to treat the $\ge$ as a $=$-sign. Here is an example:

$9x+5=42 \iff 9x=37 \iff x=4\frac19$ and replace the $=$ with $\ge$, so you get $x \ge 4\frac19$, but the main difference I think is when you divide or multiply the equation by negative numbers. When you divide or multiply the equation with a negative number, the $\ge$-sign turns into a $\le$-sign and vice versa. Here is another example:

$-6x+14\ge2 \iff -6x\ge-12$. Now you need to divide the equation by $-6$, which is negative, so you need to reverse the $\ge$-sign. You get the following equation:

$-6x\ge-12 \iff x\le2$.

The $\geq$ becomes $\leq$ if you multiply by a negative number and vice versa. Also you're looking only at linear equations there. If you have a polynomial of $\deg(2+)$ then it becomes different from solving equalities(in fact you find roots in that case too by solving the equality, but then you check for what $x$ the inequality holds).

Hint: what does it mean to solve an equation?

It means to find a solution, a set of numbers that satisfies that equation.

So when we use ≤ and ≥, we are pretty much looking for all numbers that fit that constraint. For example, if I told you to plot this:

$$y≥2x+1$$

you might be inclined to plot a line like you would with $y=2x+1$, but in reality, we want all the $x$'s and $y$'s that not only fall on this line, but above it, as well; we want all the $x$'s and $y$'s such that $y=2x+1$ and $y>2x+1$.

In other words, we want all the $x's$ and $y$'s such that $y$ is greater than or equal to $2x+1$.

• So you're saying that $\ge$ and $\le$ represent a set of numbers?
– user185734
Commented Oct 20, 2014 at 19:26
• @TheMathMan $\ge$ and $\le$ represent an relation between the numbers, just like $=$ represents an relation. The solution to the relation is the set of all numbers such that the relation is true. Commented Oct 20, 2014 at 19:34
• I hope this a good idea for me, Darksonn. Also, why does my picture keep changing?
– user185734
Commented Oct 20, 2014 at 19:40