Inequalities

$3X>6$

In this case, it is a simple inequality. Just like in the equation, we will divide both sides by $3$ and we will obtain:

$X>2$

This is actually the solution. Every $X$ is greater than $2$  

$-2X>4$

In this case, note that we must divide both sides by a negative number $-2$ Therefore, the sign must be reversed, that is, we get:

$X<-2$

This is actually the solution. Any $X$ less than $-2$

Given:

$F\left(x\right)=4x−2$

$g\left(x\right)=−3x+5$

Find when

$F\left(x\right)>g\left(x\right)$

Solution:
We replace the equations in the inequality and we get:

$4x−2>−3x+5$

Resolve the inequality:

$7X>7$

$x>1$

This means that when

$X<1,F\left(x\right)>g\left(x\right)F\left(x\right)>g\left(x\right)$

The following figure is given, in which the graphs of the two lines appear:

$F\left(x\right)=4x−2$

$g\left(x\right)=−3x+5$

The two graphs meet at the point $X=1$

According to the graph, find when $f\left(X\right)>g\left(X\right)$

Solution:
The first step:
We will identify which graph belongs to which function.

We can see it in the linear equation $F\left(x\right)$

$F\left(x\right)=4x−2$

The slope is positive - the line goes up and its intersection point with the $Y$ axis is $−2$.
Therefore, the blue graph will be $F\left(X\right)$

Furthermore,
we can see that in the linear equation $G\left(x\right)$

The slope is negative: the line goes down and its intersection point with the $Y$ axis is $5$.
Therefore, the purple graph will be

$g\left(x\right)=−3x+5$

$F\left(X\right)$

The second step:
We will write next to each graph its name.

We check when $f\left(X\right)>g\left(X\right)$ That is, at what values of $X$ is the graph of $F\left(x\right)$ greater than the graph of \( g\left(X\right) \ ?).
Let's look at the illustration in front of us, this time with the signs:

We note that we are told that the graphs meet at the point where $X=1$
We will examine the graphs and ask when $f\left(X\right)$ Is the blue graph greater than, $g\left(X\right)$  The purple graph?
The answer is when $X>!$
Pay attention, in both directions we arrive at the same answer and not by coincidence.