Graphing Inequalities Finding The Solution Set For S+l Less Than 30 And 8s+12l Less Than Or Equal To 160

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Introduction

In mathematics, understanding how to represent and solve systems of inequalities is a crucial skill. Graphing inequalities provides a visual method to identify the solution set, which includes all points that satisfy all inequalities simultaneously. This article will guide you through the process of graphing the solution set for the system of inequalities: $s + l < 30$ and $8s + 12l extless 160$. We'll break down each step, ensuring you grasp the underlying concepts and techniques. By the end of this guide, you'll be well-equipped to tackle similar problems and interpret graphical solutions effectively.

Understanding Linear Inequalities

Before diving into the specific system of inequalities, let's recap the basics of linear inequalities. A linear inequality is similar to a linear equation but uses inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). When we graph a linear inequality, we're essentially plotting all the points that satisfy the inequality. The graph is typically a region of the coordinate plane, bounded by a line. This line is solid if the inequality includes 'equal to' (≤ or ≥) and dashed if it doesn't (< or >). The region representing the solution is shaded, indicating that all points within this region satisfy the inequality.

Key Concepts in Linear Inequalities

  1. Boundary Line: The line that separates the region satisfying the inequality from the region that doesn't. This line is obtained by replacing the inequality symbol with an equal sign. For example, in $s + l < 30$, the boundary line is $s + l = 30$.
  2. Solid vs. Dashed Line: A solid line indicates that the points on the line are included in the solution (for ≤ or ≥), while a dashed line means the points on the line are not included (for < or >).
  3. Shading the Solution Region: To determine which side of the boundary line to shade, you can test a point (like (0,0) if it's not on the line) in the original inequality. If the inequality holds true, shade the side containing the test point; otherwise, shade the opposite side.

Step-by-Step Solution for the System of Inequalities

Now, let's apply these concepts to our system of inequalities:

  1. s+l<30s + l < 30

  2. 8s+12lextless1608s + 12l extless 160

We'll go through each inequality step by step to graph them and then find the common solution region.

Step 1: Graphing $s + l < 30$

First, we need to graph the boundary line for the inequality $s + l < 30$. To do this, we replace the inequality symbol with an equal sign:

s+l=30s + l = 30

This is a linear equation, and we can graph it by finding two points on the line. A convenient way to find these points is by setting $s = 0$ and then $l = 0$.

  • If $s = 0$, then $0 + l = 30$, so $l = 30$. This gives us the point (0, 30).
  • If $l = 0$, then $s + 0 = 30$, so $s = 30$. This gives us the point (30, 0).

Plot these two points (0, 30) and (30, 0) on a coordinate plane and draw a line through them. Since the inequality is $s + l < 30$, we use a dashed line to indicate that the points on the line are not included in the solution.

Next, we need to determine which side of the line to shade. We can test the point (0, 0) in the original inequality:

0+0<300 + 0 < 30

0<300 < 30

This is true, so we shade the region that contains the point (0, 0). This region is below the dashed line.

Step 2: Graphing $8s + 12l extless 160$

Now, let's graph the second inequality, $8s + 12l extless 160$. Again, we start by replacing the inequality symbol with an equal sign:

8s+12l=1608s + 12l = 160

To make the equation easier to work with, we can simplify it by dividing all terms by 4:

2s+3l=402s + 3l = 40

Now, we find two points on this line by setting $s = 0$ and then $l = 0$.

  • If $s = 0$, then $2(0) + 3l = 40$, so $3l = 40$, and $l = rac{40}{3} ext{approximately} 13.33$. This gives us the point (0, 13.33).
  • If $l = 0$, then $2s + 3(0) = 40$, so $2s = 40$, and $s = 20$. This gives us the point (20, 0).

Plot these two points (0, 13.33) and (20, 0) on the same coordinate plane. Since the inequality is $8s + 12l extless 160$, we use a dashed line to indicate that the points on the line are not included in the solution.

Next, we determine which side of the line to shade by testing the point (0, 0) in the original inequality:

8(0)+12(0)extless1608(0) + 12(0) extless 160

0extless1600 extless 160

This is true, so we shade the region that contains the point (0, 0). This region is above the dashed line.

Step 3: Finding the Solution Set

The solution set for the system of inequalities is the region where the shaded regions of both inequalities overlap. This overlapping region represents all the points that satisfy both inequalities simultaneously. On the graph, this will be the area where the shading from both inequalities intersects.

  1. Identify the Overlapping Region: Look for the area on the graph where both shaded regions intersect. This is the region that satisfies both inequalities.
  2. Mark the Solution Set: Shade this overlapping region more distinctly or use a different color to highlight it.

This overlapping region is the solution set for the system of inequalities. Any point within this region, but not on the dashed boundary lines, is a solution to the system.

Common Mistakes to Avoid

When graphing systems of inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

  1. Using a Solid Line Instead of a Dashed Line (or Vice Versa): Remember that a solid line is used when the inequality includes 'equal to' (≤ or ≥), and a dashed line is used when it doesn't (< or >). Incorrectly drawing the line can change the solution set.

  2. Shading the Wrong Region: Always test a point (like (0,0)) to determine which side of the line to shade. If the inequality is true for the test point, shade the region containing that point; otherwise, shade the opposite region.

  3. Incorrectly Simplifying the Inequality: Ensure you simplify the inequality correctly before graphing. For example, dividing all terms by a common factor can make the equation easier to work with, but make sure you do it accurately.

  4. Failing to Identify the Overlapping Region: The solution to the system is the region where the shaded areas of all inequalities overlap. Be sure to identify and clearly mark this region.

  5. Not Understanding the Context: Always consider the context of the problem. For example, if the variables represent real-world quantities like the number of items, negative values may not be meaningful.

Real-World Applications

Understanding how to graph and solve systems of inequalities has numerous real-world applications. These skills are particularly useful in fields such as economics, business, and engineering, where constraints and limitations often need to be considered.

Business and Economics

In business, systems of inequalities can be used to model constraints on resources, such as budget and labor, and to optimize production or profit. For example, a company might use inequalities to determine the optimal combination of products to manufacture given limitations on raw materials, labor hours, and market demand.

Engineering

Engineers often use systems of inequalities to design structures and systems within certain specifications and constraints. For instance, they might use inequalities to ensure that a bridge can support a certain weight or that a chemical process operates within safe temperature and pressure ranges.

Resource Allocation

Systems of inequalities can also be used to allocate resources efficiently. For example, a city planner might use inequalities to determine how to distribute funds among different projects while staying within budget constraints and meeting various community needs.

Optimization Problems

Many optimization problems involve maximizing or minimizing a certain quantity subject to a set of constraints. These constraints can often be expressed as inequalities, and graphing these inequalities can help visualize the feasible region and find the optimal solution.

Conclusion

Graphing the solution set for systems of inequalities, such as $s + l < 30$ and $8s + 12l extless 160$, is a fundamental skill in mathematics with wide-ranging applications. By understanding the basics of linear inequalities, including how to graph boundary lines and shade solution regions, you can effectively solve these systems. Remember to avoid common mistakes, such as using the wrong type of line or shading the incorrect region. With practice, you'll become proficient in visually representing and interpreting the solutions to systems of inequalities, making you well-prepared to tackle more complex mathematical problems and real-world applications.

By following the step-by-step guide and tips provided in this article, you should now have a solid understanding of how to graph and solve systems of inequalities. Keep practicing, and you'll be able to apply these skills confidently in various contexts.