Graphing Linear Inequalities

in StemSocial2 months ago (edited)

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.


Pixabay Image Source

 

Topics


 

  • Review Of Linear Equations
  • Isolate For y First
  • Graph The Inequalities

 

Review Of Linear Equations


The equation for a linear function is of the following form:

 

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.

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Isolate For y First


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.

 

Example One

Solve for y in 2y - 6 > x.

 

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

 

Example Two

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

 

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

 

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.

 

Graph The Inequalities


Once y is isolated, the graphing can commence. Here are some examples.

 

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.

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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.

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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.

 

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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.

 

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:

 

 

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.

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Thank you for reading.

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