Solving System Of Inequalities: A Step-by-Step Guide

Hey guys! Today, we're diving into solving a system of inequalities. Specifically, we'll tackle the system:

$3y \leq -2x + 18$

$-4y \leq -x + 12$

Solving systems of inequalities might seem daunting at first, but with a step-by-step approach, it becomes quite manageable. We'll break down each inequality, graph them, and then find the region that satisfies both. So, let's get started!

Step 1: Simplify the Inequalities

First, we need to get each inequality into a more manageable form. Ideally, we want to isolate 'y' on one side. This makes it easier to graph the inequalities later on. So, let's take each inequality one by one.

Inequality 1: $3y \leq -2x + 18$

To isolate 'y', we simply divide both sides of the inequality by 3:

$y \leq \frac{-2}{3}x + 6$

This form is now much easier to work with. We can clearly see the slope ($-\frac{2}{3}$) and the y-intercept (6).

Inequality 2: $-4y \leq -x + 12$

Here, we need to be a bit more careful because we're dividing by a negative number (-4). Remember, when you multiply or divide an inequality by a negative number, you need to flip the inequality sign.

So, dividing both sides by -4, we get:

$y \geq \frac{1}{4}x - 3$

Notice that the $\leq$ sign has changed to $\geq$. This is crucial for getting the correct solution. Again, we can easily identify the slope ($\frac{1}{4}$) and the y-intercept (-3).

Now that we have both inequalities in the slope-intercept form, we can move on to graphing them.

Step 2: Graphing the Inequalities

Graphing inequalities involves plotting the lines and then shading the region that satisfies the inequality. Let's graph each inequality separately and then combine them to find the solution region.

Graphing $y \leq \frac{-2}{3}x + 6$

1. Plot the line: First, treat the inequality as an equation: $y = \frac{-2}{3}x + 6$. Start by plotting the y-intercept, which is (0, 6). Then, use the slope to find another point. The slope is $-\frac{2}{3}$, which means for every 3 units you move to the right, you move 2 units down. So, another point on the line is (3, 4). Draw a straight line through these two points. Since the inequality is $y \leq \frac{-2}{3}x + 6$, the line should be solid, indicating that points on the line are included in the solution.
2. Shade the region: Since we want $y \leq \frac{-2}{3}x + 6$, we need to shade the region below the line. This is because all the y-values in that region are less than or equal to the corresponding y-values on the line.

Graphing $y \geq \frac{1}{4}x - 3$

1. Plot the line: Treat the inequality as an equation: $y = \frac{1}{4}x - 3$. Start by plotting the y-intercept, which is (0, -3). The slope is $\frac{1}{4}$, which means for every 4 units you move to the right, you move 1 unit up. So, another point on the line is (4, -2). Draw a straight line through these two points. Since the inequality is $y \geq \frac{1}{4}x - 3$, the line should also be solid.
2. Shade the region: Since we want $y \geq \frac{1}{4}x - 3$, we need to shade the region above the line. This is because all the y-values in that region are greater than or equal to the corresponding y-values on the line.

Combining the Graphs

Now, we need to combine the two graphs to find the region that satisfies both inequalities. This is the region where the shaded areas of both inequalities overlap. The overlapping region represents all the points (x, y) that satisfy both $y \leq \frac{-2}{3}x + 6$ and $y \geq \frac{1}{4}x - 3$.

Step 3: Identifying the Solution Region

The solution region is the area where the shaded regions of both inequalities overlap. This region is bounded by the two lines we graphed. Any point within this region, or on the solid lines, is a solution to the system of inequalities. To verify, you can pick a point in the overlapping region and plug its coordinates into both original inequalities. If both inequalities hold true, then that point is indeed a solution.

Example Verification

Let's pick the point (0, 0) and see if it satisfies both inequalities:

1. $3y \leq -2x + 18$

   $3(0) \leq -2(0) + 18$

   $0 \leq 18$ (True)

2. $-4y \leq -x + 12$

   $-4(0) \leq -(0) + 12$

   $0 \leq 12$ (True)

Since (0, 0) satisfies both inequalities, it lies within the solution region, which confirms our graphical solution.

Step 4: Writing the Solution Set

The solution to the system of inequalities is the set of all points (x, y) that lie within the overlapping region, including the points on the boundary lines. Formally, we can describe the solution set as:

$\left\{(x, y) \mid 3y \leq -2x + 18 \text{ and } -4y \leq -x + 12\right\}$

This notation simply means the solution set consists of all ordered pairs (x, y) such that both inequalities are true.

Tips and Tricks for Solving Systems of Inequalities

• Always double-check the inequality sign when multiplying or dividing by a negative number. This is a common mistake that can lead to an incorrect solution.
• Use test points to verify your solution region. Pick a point within the overlapping region and plug its coordinates into the original inequalities to make sure they hold true.
• If the lines are parallel, the system may have no solution or infinitely many solutions. If the shaded regions do not overlap, there is no solution. If the shaded regions completely overlap, then all points on or between the lines are solutions.
• Practice makes perfect! The more you practice solving systems of inequalities, the more comfortable and confident you'll become.

Common Mistakes to Avoid

• Forgetting to flip the inequality sign: As mentioned earlier, this is a very common mistake. Always remember to flip the inequality sign when multiplying or dividing by a negative number.
• Incorrectly shading the region: Make sure you're shading the correct region based on the inequality sign. If the inequality is $y \leq$, shade below the line. If the inequality is $y \geq$, shade above the line.
• Using a dashed line instead of a solid line (or vice versa) when graphing.: Remember, if the inequality includes an "equal to" component ($y \leq$ or $y \geq$), use a solid line to indicate that points on the line are included in the solution. If the inequality does not include an "equal to" component ($y <$ or $y >$), use a dashed line to indicate that points on the line are not included in the solution.
• Not checking the solution: Always verify your solution by plugging a point from the overlapping region into the original inequalities.

Conclusion

Solving systems of inequalities involves simplifying the inequalities, graphing them, and identifying the region that satisfies all the inequalities. By following these steps and avoiding common mistakes, you can confidently solve any system of inequalities. Remember to practice regularly and double-check your work to ensure accuracy. Keep up the great work, and you'll master this topic in no time! Hopefully, this guide helped you understand the process. Good luck, and happy solving! Solving inequalities can sometimes be frustrating, but with a little practice, you'll get the hang of it. Keep practicing, and you'll become an expert in no time!