Algebra 1 - Graphing Inequalities in Two Variables

Summary

This lesson teaches how to graph inequalities on a coordinate plane with two variables, explaining the difference between dashed and solid lines based on the inequality symbols. It covers concepts like shading regions above or below the line depending on the inequality's direction (greater than or less than). The tutorial highlights how to graph inequalities such as 'y < x' or 'y ≥ x,' and how to determine which points are solutions. The video provides step-by-step examples, helping students grasp the visualization of inequalities and the rules for shading and identifying solutions in the coordinate plane.

Takeaways

• 😀 Dashed lines represent inequalities without an equal sign (e.g., '<' or '>'), while solid lines represent inequalities with an equal sign (e.g., '<=' or '>=').
• 😀 For inequalities like 'y < x' or 'y > x', the graphing process involves plotting the line and then checking if the inequality symbol requires shading above or below the line.
• 😀 When graphing inequalities, the first step is to treat them like an equation (e.g., y = x) and plot the line, but then stop to assess whether the line should be dashed or solid based on the inequality symbol.
• 😀 Shading for inequalities depends on whether the y-values are greater or less. For example, for 'y < x', you shade below the line, and for 'y > x', you shade above it.
• 😀 Points on a dashed line are not part of the solution set. Points on a solid line **are** part of the solution set.
• 😀 When shading inequalities, always check whether the values of the variables satisfy the inequality and whether they are above or below the line, depending on the inequality symbol.
• 😀 When working with inequalities like 'y <= 3x - 2', ensure to plot the y-intercept and apply the slope to plot the line, then determine whether the shading should be above or below the line.
• 😀 For inequalities involving x (e.g., 'x >= -2'), evaluate the x-axis values and shade the correct side of the graph based on whether x-values are greater or smaller than the specified value.
• 😀 The solution set includes all points that satisfy the inequality, and for inequalities with 'equal to' symbols, the points directly on the line are part of the solution.
• 😀 Understanding the difference between dashed and solid lines and how to determine which region to shade makes graphing inequalities straightforward, whether for simple equations or more complex systems.

Q & A

What is the primary difference between graphing inequalities on a number line and on a coordinate plane?

On a number line, you use open or closed circles to represent inequalities, while on a coordinate plane, these are represented using dashed or solid lines to indicate whether the boundary is included in the solution set.

How do you graph the inequality y < x on the coordinate plane?

First, graph the line y = x as if it were an equation. Since the inequality is 'less than', draw a dashed line (indicating the boundary is not included), and then shade below the line because the inequality represents values less than x.

What does a dashed line represent when graphing inequalities on a coordinate plane?

A dashed line represents inequalities like 'y < x' or 'y > x' where the boundary is not part of the solution set, meaning points on the dashed line are not included in the solution.

When graphing y ≤ x, what kind of line should you draw and why?

You should draw a solid line because the inequality includes 'less than or equal to', meaning the points on the line are part of the solution set.

What does shading on a graph represent when graphing inequalities?

Shading on a graph represents the region of the coordinate plane that satisfies the inequality. For 'y < x', you would shade below the line, and for 'y > x', you would shade above the line.

How do you determine which side of the line to shade when graphing an inequality?

You look at the inequality symbol. If it’s 'less than' or 'less than or equal to', you shade the side where the y-values are smaller (below the line). If it’s 'greater than' or 'greater than or equal to', you shade the side where the y-values are larger (above the line).

Why is the point (0, 0) not part of the solution for y < x, even though it lies on the dashed line?

Since the inequality is strict ('<'), the points on the dashed line are not included in the solution set. Therefore, (0, 0), which lies on the line y = x, is not part of the solution.

What should you do if you encounter a point directly on the solid line when graphing an inequality like y ≤ x?

Points on the solid line are included in the solution set for inequalities with '≤' or '≥'. So, if a point lies on the solid line, it is part of the solution.

How do you graph inequalities that involve complex expressions like y < 3x - 2?

First, graph the line y = 3x - 2 as an equation (plotting the y-intercept and using the slope). Since the inequality is 'less than', draw a dashed line and shade below it because y-values are less than the values on the line.

What does it mean to graph an inequality in standard form, such as 2x + 3y < 6?

To graph an inequality in standard form, convert it to slope-intercept form if necessary, or find the intercepts directly. Then, graph the boundary line and use the inequality symbol to determine whether it’s a dashed or solid line, followed by shading the correct region of the plane.