Problem 23 - Algebra 1

Question

Graph the system of linear inequalities. \(-\frac{3}{2} x+y \leq 3\) \(\frac{1}{4} x+y>-\frac{1}{2}\) \(4 x+y<2\)

Step-by-Step Solution

Answer

The graphed system of inequalities shows the lines and shaded regions for each individual inequality along with the overlapped region that represents the solution to the system. The lines are drawn as solid or dashed based on whether the inequality includes 'equal to'. The shading direction for each inequality is determined by choosing a test point.

Step 1: Graph the first inequality

For the first inequality, \(-\frac{3}{2} x+y \leq 3\), start by treating it as an equality and plot the line \(-\frac{3}{2} x+y = 3\). This line should be solid because the inequality includes 'equal to'. To shade the region for 'less than or equal to', a good strategy is to pick a test point not on the line, like the origin (0,0). Substitute these values into the original inequality. If it satisfies, then shade the region that includes this test point.

Step 2: Graph the second inequality

For the second inequality, \(\frac{1}{4} x+y > -\frac{1}{2}\), again treat it as an equality first to plot the line \(\frac{1}{4} x+y = -\frac{1}{2}\). This line should be dashed because the inequality doesn't include 'equal to'. To determine which side of the line to shade, use the same strategy of picking a test point.

Step 3: Graph the third inequality

For the third inequality, \(4x+y < 2\), plot the line for \(4x+y = 2\). This line should also be dashed. Use a test point to decide which side of the line to shade.

Step 4: Find the solution to the system of inequalities

The solution to the system of inequalities is the region where all of the shaded areas from the three inequalities overlap.

Key Concepts

System of Inequalities

When working with a system of inequalities, you are dealing with two or more inequalities at the same time. These inequalities share a common set of variables, and the solution consists of all the ordered pairs that satisfy all inequalities in the system simultaneously. It's like looking for a common ground where all the conditions are met.

• Systematic Approach: Always start by graphing each inequality one by one.
• Boundary Lines: Draw a solid line if the inequality includes the boundary (equal to), and a dashed line if it does not.
• Intersection: The solution region is the intersection where the shaded regions of all individual inequalities overlap.

When you graph a system, think of it as layering each individual inequality on the same plane. The collective shaded area reveals the solution set.

Shading Inequalities

The process of shading inequalities helps you visualize the solution set of an inequality. This technique indicates which side of the boundary line contains solutions that satisfy the inequality.

• Less Than: If you have an inequality with '<', shade below the boundary line (for y-inequalities).
• Greater Than: If the inequality is '>', shade above the boundary line (for y-inequalities).

Always remember the rule of thumb: A solid line means the boundary is included in the solution (' egas≤' or '≥'), while a dashed line means it is not ('<' or '>'). Your shading direction will provide a visual cue for which half-plane satisfies the inequality.

Linear Inequality Graphing

The core of linear inequality graphing lies in understanding how to represent the inequality on a coordinate plane. Here's a breakdown for a straight-forward graphing process:

• Boundary Line: Start by graphing the boundary line, which is the associated linear equation obtained by replacing the inequality sign with an equals sign.
• Line Type: Choose either a solid or dashed line to represent whether the points on the line are included (solid) or excluded (dashed) from the solution set.
• Shading: Use shading to show all the points that satisfy the inequality. This represents one half of the plane that is either above or below (or to the right or left for x-inequalities) the boundary line.

With careful graphing and clear demarcation of shaded regions, you'll be able to accurately convey the solutions to the inequality.

Test Point Method

The test point method is a foolproof way to determine which side of the boundary line to shade when graphing an inequality. Following these steps makes it clear:

• Choosing Test Points: Select simple points, such as (0,0) if it's not on the boundary line, as test points.
• Substitution: Plug the coordinates of the test point into the inequality.
• Judgment: If the inequality holds true, the side of the line containing the test point is shaded. If not, shade the opposite side.

Through this method, you eliminate guesswork and ensure accuracy in representing the solution set of the inequality. It's a quick check that helps you secure the right shaded area.