# Inequalities

Inequalities are the "outliers" of equations and many of the rules that apply to equations also apply to inequalities.
In terms of writing, the main difference is that instead of the equal sign $"="$, we use greater than $">"$ or less than $"<"$ signs.

Inequalities can be simple or more complex and also contain fractions, parentheses, and more.

Another thing that distinguishes inequalities from equations is that equations with one variable have a unique solution. On the contrary, inequalities have a range of solutions.

Inequalities between linear functions will translate into questions like when $F\left(x\right)>G\left(x\right)$ or vice versa.
We can answer this type of questions in two ways:

• Using equations
if the equations of the two functions are given, we will place them in the inequality, solve it, and find the corresponding $X$ values.
• Using graphs
we will examine at what $X$ values, $Y$ values of the function in question are higher or lower than the function in the inequality.

## Simple Inequality Instructions

### Example 1

$3X>6$

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 $2$

### Example 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 $-2$

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## Solving inequalities using equations:

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

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

## Solution of inequality using graphs:

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.

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