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GRAPHING LINEAR EQUATIONS IN TWO VARIABLES To graph a linear inequality with one variable we use the number line. To graph a linear inequality with two variables we use a two-dimensional graph. As with linear equations, linear inequalities can be written in standard form or slope intercept form. Standard Form: Ax + By < C or Ax + By > C Ax + By ≤ C or Ax + By ≥ C y < mx + b or y > mx + b y ≤ mx + b or y ≥ mx + b Slope-Intercept Form: To graph a linear inequality with two variables it is probably easier and recommended to change it to slope intercept form. There are two important rules to remember when graphing a linear inequality. If the inequality sign is > or <, the line is dashed. If the inequality is ≥ or ≤ , then the line is solid. Also, for inequalities that are greater than > or greater or equal to ≥ the graph needs to be shaded above the line. For inequalities that are less than < or less or equal to ≤ the graph needs to be shaded below the line. Examples: Graph the solutions of 3 x + 2 y ≥ 6 : Step 1 – Since the equation is in standard form, change it to slope intercept form. 3x + 2 y ≥ 6 → 2 y ≥ −3 x + 6 −3 x + 6 y≥ 2 3 y ≥ − x+3 2 ← Slope-Intercept Form Step 2 – Graph the slope-intercept form equation. Since the inequality has the sign ≥ the line is solid. Step 3 – Since the equation has the sign ≥ , the graph is shaded above the line. 3 ← The shaded area in the graph is the graphic answer to the inequality. It means that any point above the line, including the line, satisfies the inequality and can be a possible answer. Graph the solutions of x − 3 y < 4 Step 1 – Since the equation is in standard form, change it to slope intercept form. x − 3y < 4 → − 3y < −x + 4 −x + 4 y> −3 Remember to switch the inequality sign when dividing by a negative. y> 1 4 x− 3 3 ← Slope-Intercept Form Step 2 – Graph the slope-intercept form equation. Since the inequality has the sign > the line is dashed. Step 3 – Since the equation has the sign >, the graph is shaded above the line. 4 Solving a System of Linear Inequalities To solve a system of linear inequalities we need to do it graphically since the solution to a system of linear inequalities is the set of points whose coordinates satisfy all the inequalities in the system, or where both shaded areas overlap. Graph the lines as previously shown. 1 y <− x+2 Example: Determine the solution to the following system of inequalities 2 x− y ≤4 Step 1 – Change the second inequality to slope-intercept form. x− y≤4 x− x− y ≤ 4− x − y ≤ −x + 4 y ≥ x−4 ← Remember to switch the inequality sign since we are dividing by a negative number. Step 2 – Graph both inequalities in slope-intercept form and shade the solution for each inequality. Since the first inequality has the sign < the line is dashed. The second inequality has the sign ≥ so the line is solid. y ≥ x−4 Solution to the system of inequalities y<− 5 1 x+2 2 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES – EXERCISES Graph each inequality: 1 1. y ≥ − x 2 2. y < 2 x + 1 3. 10 ≥ 5 x − 2 y Determine the solution to each system of inequalities: 2x − y < 4 y ≥ −x + 2 5. y < 3x − 4 6x ≥ 2 y + 8 x≥0 1 1 7. x + y ≥ 2 2 2 2 x − 3 y ≤ −6 x≥0 y≥0 8. 2x + 3y ≤ 6 4x + y ≤ 4 x≥0 y≥0 9. 5 x + 4 y ≤ 16 x + 6 y ≤ 18 4. 6. − 3 x + 2 y ≥ −5 y ≤ −4 x + 7 x≥0 y≥0 10. x ≤ 15 30 x + 25 y ≤ 750 10 x + 40 y ≤ 800 6 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES – ANSWERS TO EXERCISES 1. 2. 3. 4. 5. 6. 7. 8. 7 9. 10. 8