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## Graphing Linear Inequalities

This is a graph of a linear inequality:

The inequality y ≤ x + 2

You can see the y = x + 2 line, and the shaded area is where y is less than or equal to x + 2

Linear Inequality

A Linear Inequality is like a Linear Equation (such as y = 2x+1) …

… but it will have an Inequality like <, >, ≤, or ≥ instead of an =.

## How to Graph a Linear Inequality

First, graph the “equals” line, then shade in the correct area.

There are three steps:

Rearrange the equation so “y” is on the left and everything else on the right.

Plot the “y=” line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>)

Shade above the line for a “greater than” (y> or y≥) or below the line for a “less than” (y< or y≤).

Let us try some examples:

Example: y≤2x-1

1. The inequality already has “y” on the left and everything else on the right, so no need to rearrange

2. Plot y=2x-1 (as a solid line because y≤ includes equal to)

3. Shade the area below (because y is less than or equal to)

Example: 2y − x ≤ 6

1. We will need to rearrange this one so “y” is on its own on the left:

Start with: 2y − x ≤ 6

Add x to both sides: 2y ≤ x + 6

Divide all by 2: y ≤ x/2 + 3

2. Now plot y = x/2 + 3 (as a solid line because y≤ includes equal to)

3. Shade the area below (because y is less than or equal to)

Example: y/2 + 2 > x

1. We will need to rearrange this one so “y” is on its own on the left:

Start with: y/2 + 2 > x

Subtract 2 from both sides: y/2 > x − 2

Multiply all by 2: y > 2x − 4

2. Now plot y = 2x − 4 (as a dashed line because y> does not include equals to)

3. Shade the area above (because y is greater than)

2x – 4 “>

The dashed line shows that the inequality does not include the line y=2x-4.

## Two Special Cases

You could also have a horizontal or vertical line:

=1″>

This shows where y is less than 4 (from, but not including, the line y=4 on down)Notice that we have a dashed line to show that it does not include where y=4

This one doesn’t even have y in it!It has the line x=1, and is shaded for all values of x greater than (or equal to) 1

Graphing inequalities (x-y plane) review (article) | Khan – We graph inequalities like we graph equations but with an extra step of shading one side of the line. This article goes over examples and gives you a chance to practice. Modeling with linear inequalities. (x-y plane) review. This is the currently selected item. Next lesson. Modeling with linear inequalities. Sort by:First we are going to find the equation of the solid line passing trough the points (0, 3) and (2, 4). Using the slope formula: Now we can use the point slope formula to complete the line equation: Since the shaded region is bellow the line , the inequality represented in the graph is .An ordered pair \((x,y)\) is a solution to a linear inequality if the inequality is true when we substitute the values of x and y. Example \(\PageIndex{1}\) Determine whether each ordered pair is a solution to the inequality y>x+4:

Which linear inequality is represented by the graph? y – inequality, and is best represented by its graph. The graph of a linear inequality is represented by a straight or dashed line and a shaded half-plane. An illustration is shown below. Example 1: Without graphing, determine whether ( 3, 7)− − is a solution to y x> − 4. Solution: Substitute x = − 3 and y = − 7 into the inequality andGraph the line y = 2x-4 as a dashed line. After you have plotted the line, shade in the area to right of the line. To help you to graph the line, two points are (0,-4) and (2,0) Remember to make it a dashed line to indicate that it is not included. Here is a how it should look:1. Graph the solution to y < 2x + 3. 2. Graph the inequality: 4(x + y) – 5(2x + y) < 6 and answer the questions below. a. Check whether the point (-22, 10) is within the solution set. b. Determine the slope of the border line. 3. Graph the inequality of y< 3x and determine which quadrant will be completely shaded. 4.

4.1: Graphing Linear Inequalities in Two Variables – Graph 5x-y=4. Solve for . Tap for more steps… Subtract from both sides of the equation. Multiply each term in by . Tap for more steps… Multiply each term in by . To find the x-intercept (s), substitute in for and solve for . Solve the equation. Tap for more steps… Rewrite the equation as .The line represents the inequality y .Hence, is correct. Further explanation: The linear equation with slope m and intercept c is given as follows. The formula for slope of line with points and can be expressed as, . Given:See the explanation below. Graph: y>=-3x+4 Determine several points on the line by choosing values for x and solving for y. Example: "Points" x=1,y=1 x=0,y=4 x=-1,y=7 Plot your points on a graph and draw a solid straight line through the points. Since the inequality is >, shade the area above the line. graph{y>=-3x+4 [-16.22, 15.8, -6.54, 9.48]}