# Linear Inequalities in Two Variables (Part 1)

Need some basic understanding of sentences in two variables first? Introduction to Equations and Inequalities in Two Variables

Here's what this lesson offers:

## Going from Linear Equations to Linear Inequalities

You've already learned that the graph of $\,y = x + 1\,$ is the line shown below. This line is the picture of all the points $\,(x,y)\,$ that make the equation ‘$\,y=x+1\,$’ true.

How can ‘$\,y=x+1\,$’ be true? For a given $\,x$-value, the $\,y$-value must equal $\,x + 1\,.$ For each $\,x$-value, there is exactly one corresponding $\,y$-value—whatever $\,x\,$ is, plus $\,1\,.$

The line is the picture of all the points $\,(x,x+1)\,,$ as $\,x\,$ varies over all real numbers.

The line ‘$\,y=x+1\,$’ is all points of the form:

$$\bigl( \cssId{s16}{x\ \ ,} \cssId{s17}{\overbrace{x+1}^{\text{the y-value EQUALS x+1}}} \bigr)$$

Graph of $\,y = x + 1\,$:
all points of the form $\,(x,x+1)$

Question: What happens if the verb in the sentence ‘$\,y \color{red}{=} x+1\,$’ is changed from ‘$\,\color{red}{=}\,$’ to $\,\lt\,,$  $\,\gt\,,$  $\,\le\,,$  or  $\,\ge\,$?

Answer: You go from a linear equation in two variables, to a linear inequality in two variables. The solution set changes dramatically! What was a line now becomes an entire half-plane:

Graph of $\,y < x + 1\,$:

All points of the form $\,(x,y)\,$ where the $y$-value is less than $\,x+1\,.$

Line is dashed; shade below the line.

Graph of $\,y > x + 1\,$:

All points of the form $\,(x,y)\,$ where the $y$-value is greater than $\,x+1\,.$

Line is dashed; shade above the line.

Graph of $\,y \le x + 1\,$:

All points of the form $\,(x,y)\,$ where the $y$-value is less than or equal to $\,x+1\,.$

Line is solid; also shade below the line.

Graph of $\,y \ge x + 1\,$:

All points of the form $\,(x,y)$ where the $y$-value is greater than or equal to $\,x+1\,.$

Line is solid; also shade above the line.

## Important Concepts for Graphing Linear Inequalities in Two Variables

### Definition: Linear Inequality in Two Variables

A linear inequality in two variables is a sentence of the form $$\cssId{s41}{ax + by + c < 0}\,,$$ where $\,a\,$ and $\,b\,$ are not both zero; $\,c\,$ can be any real number.

The inequality symbol can be any of these: $$\cssId{s45}{\lt\,,\ \ \gt\,,\ \ \le\,,\ \ \ge}$$

Remember:   ‘A sentence of the form ...’   really means   ‘A sentence that can be put in the form ...’

### Examples of Linear Inequalities in Two Variables:

$$\begin{gather} \cssId{s51}{3x - 4y + 5 \gt 0}\cr \cr \cssId{s52}{y \le 5x - 1}\cr \cr \cssId{s53}{x \ge 2}\cr \cssId{s54}{\text{(shorthand for \,x + 0y \ge 2\,)}}\cr \cr \cssId{s55}{y \lt 5}\cr \cssId{s56}{\text{(shorthand for \,0x + y \lt 5\,)}} \end{gather}$$

Key ideas for recognizing linear inequalities in two variables:

• The verb must be an inequality symbol: $$\cssId{s59}{\lt\,,\ \ \le\,,\ \ \gt\,,\ \ \text{ or } \ge}$$
• The variables must be raised only to the first power: no squares, no variables in denominators, no variables under square roots, and so on.
• You don't need to have both $\,x\,$ and $\,y\,,$ but you must have at least one of these variables.

### Linear Inequalities Graph as Half-Planes:

Every linear inequality in two variables graphs as a half-plane:

• If the verb is $\,\lt\,$ or $\,\gt\,,$ the boundary line is not included (dashed).
• If the verb is $\,\le\,$ or $\,\ge\,,$ the boundary line is included (solid).

### Which Half-Plane to Shade?

If the linear inequality is in slope-intercept form (like $\,y \lt mx + b\,$), then it's easy to know which half of the line to shade:

• If the sentence is  $\,y \lt mx + b\,$  or  $\,y \le mx + b\,,$ shade below the line.
• If the sentence is  $\,y \gt mx + b\,$  or  $\,y \ge mx + b\,,$ shade above the line.

This only works if the inequality is in slope-intercept form! Of course, you can always put a sentence in slope-intercept form, by solving for $\,y\,.$ Then, you can use this method. But, the ‘test point method’ (in Part 2) is usually quicker-and-easier, if the sentence isn't already in slope-intercept form.