Hi there. In this mathematics based post, I cover graphing linear inequalities. It is assumed that the reader is familiar with linear equations and the algebra associated with linear functions.

As usual, I use QuickLatex.com for math text rendering.

<center><img src="https://cdn.pixabay.com/photo/2016/03/03/10/20/mathematics-1233876_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2016/03/03/10/20/mathematics-1233876_960_720.jpg">Pixabay Image Source</a></center>

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## Topics
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* Review Of Linear Equations
* Isolate For y First
* Graph The Inequalities

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## Review Of Linear Equations
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The equation for a linear function is of the following form:

<center>https://quicklatex.com/cache3/bc/ql_43f37d08451711818d2444f5c31764bc_l3.png</center>

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The variable `y` is the dependent variable that depends on the value of the independent variable `x`. The value of `b` is the y-intercept or the value of `y` when `x` is 0. To represent the change in `y` for every one unit increase of `x`, we have the slope value `m`. 

An example of a linear function is `y = 10x + 20` . I show a screenshot graph below.

![n0pen1tunk6zxtxc6s5h](https://res.cloudinary.com/drrz8xekm/image/upload/v1614873393/n0pen1tunk6zxtxc6s5h.png) 

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## Isolate For y First
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Not all equations will have the dependent variable `y` already isolated. This is where algebra comes into play. Isolating for `y` is the key step before graphing the linear inequality.

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**Example One**

Solve for y in `2y - 6 > x`.

<center>https://quicklatex.com/cache3/3b/ql_db3792e02e7da81dc4d428e4a42af13b_l3.png</center>

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The slope for this linear function is one half. Three is the y-intercept here.


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**Example Two**

Isolate for y in `-2x + 5y < 10`.

<center>https://quicklatex.com/cache3/e7/ql_29fcd5b71695bc4eda08b59beef22be7_l3.png</center>

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With this linear function, the y-intercept is 2 and the slope is two fifths.

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**Example Three**

Isolate for y in `-10y \geq 200`.

When it comes to multiplying or dividing by a negative sign, do make sure to change the direction of the inequality sign.

<center>https://quicklatex.com/cache3/7a/ql_c32a05b4db8b85b9a8cbc97eedfd027a_l3.png</center>

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## Graph The Inequalities
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Once y is isolated, the graphing can commence. Here are some examples.

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**Example One**

In example one from the previous section we obtained `y > x/2 + 3`.

On graphing paper, graph the line `y > x/2 + 3` first. The line should be dashed instead of a solid line. This is because we have a strict inequality. If it was greater or equal to with this symbol ≥ then the line can be solid. As `y` is greater than the line, the shaded region is (vertically) above the line. In the Desmos screenshot below, the shaded region above the dashed line is red.

![svlnxvjgttyfwyfrnbat](https://res.cloudinary.com/drrz8xekm/image/upload/v1614873580/svlnxvjgttyfwyfrnbat.png) 

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**Example Two**

What would the graph look like for `-2x + 5y < 10` back in example two from the previous section?

Isolating for `y` yielded `y < 2x/5 + 2`.

The line `y < 2x/5 + 2` is graphed first. As the inequality sign is less than (<), the line is dashed. In addition, the region below the line is shaded. 

![revs5nik84yh5u6anm6u](https://res.cloudinary.com/drrz8xekm/image/upload/v1614873639/revs5nik84yh5u6anm6u.png) 


**Example Three**

Graph the inequality `-2y + 9x ≥ -2`.

With this do be careful with the negative sign. Anytime you multiply or divide by a negative number, do change the direction of the inequality sign.

<center>https://quicklatex.com/cache3/ed/ql_e39bff8832ce8e9d05c651f67e21e7ed_l3.png</center>

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![f43ahrjzpfbwgjryzlbk](https://res.cloudinary.com/drrz8xekm/image/upload/v1614874341/f43ahrjzpfbwgjryzlbk.png) 

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**Example Four - Phone Plan Charges**

Suppose you buy a phone outright at its full costs. The cost for this new phone is about $2700 CAD. To supplement the cost of this phone, the phone plan is a $80 CAD phone plan per month. Keeping overage fees in mind, what would be total costs paid over time look like with a graph? The shaded region would represents total costs paid if overage fees occurred.

<center>https://quicklatex.com/cache3/1d/ql_92f3f9fd2496dec723c33276143d031d_l3.png</center>

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where `n` is the number of months paid and `P(x)` is the total amount paid to the phone company at month `n`.

As we may incur extra fees, we have this inequality:

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<center>https://quicklatex.com/cache3/fe/ql_f4cd29a205b0502681f2a0d8329747fe_l3.png</center>

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The Desmos screenshot of this does not look great. Having a large slope of 80 makes the line steep to the point that it is almost vertical. The shaded region is above the dashed line is coloured blue.

![e4t77xsuzclsqbutvuhd](https://res.cloudinary.com/drrz8xekm/image/upload/v1614874318/e4t77xsuzclsqbutvuhd.png) 

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<center>Thank you for reading.</center>