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.

## 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.

## 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.

**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.

**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.

**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.