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

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Hey guys! Today, we're diving into the exciting world of solving systems of inequalities. Don't worry, it's not as intimidating as it sounds. We'll break it down step by step, making it super easy to follow. Our example system is:

2x+y⩾4−5x+2y<1\begin{aligned} 2x + y \geqslant 4 \\ -5x + 2y < 1 \end{aligned}

Let's get started!

Understanding the Basics

Before we jump into solving, let's quickly recap what inequalities are. Unlike equations that have definite solutions, inequalities represent a range of possible values. When we have a system of inequalities, we're looking for the region where all the inequalities are true simultaneously. This region is often represented graphically.

Think of it like finding the sweet spot that satisfies multiple conditions at the same time. For instance, you might want to find all the combinations of hours studying and hours sleeping that result in both good grades and feeling rested. That's the power of inequalities!

Step 1: Isolating 'y' in Each Inequality

Our first goal is to rewrite each inequality so that 'y' is isolated on one side. This makes it easier to graph the inequalities. Let's start with the first inequality:

2x+y⩾42x + y \geqslant 4

To isolate 'y', we subtract '2x' from both sides:

y⩾−2x+4y \geqslant -2x + 4

Now, let's do the same for the second inequality:

−5x+2y<1-5x + 2y < 1

First, add '5x' to both sides:

2y<5x+12y < 5x + 1

Next, divide both sides by '2':

y<52x+12y < \frac{5}{2}x + \frac{1}{2}

So, our system now looks like this:

y⩾−2x+4y<52x+12\begin{aligned} y \geqslant -2x + 4 \\ y < \frac{5}{2}x + \frac{1}{2} \end{aligned}

Step 2: Graphing the Inequalities

Now comes the fun part – graphing! Each inequality represents a region on the coordinate plane. The boundary of each region is a line. Here's how to graph each inequality:

Graphing $y \geqslant -2x + 4$

  1. Draw the line: Graph the line $y = -2x + 4$. This is a line with a slope of -2 and a y-intercept of 4. To draw it, you can plot the y-intercept (0, 4) and use the slope to find another point. For example, go 1 unit to the right and 2 units down to find the point (1, 2).
  2. Solid or dashed line? Since the inequality is $y \geqslant -2x + 4$, it includes the 'equal to' part. This means the line is solid. A solid line indicates that the points on the line are part of the solution.
  3. Shade the region: The inequality is $y \geqslant -2x + 4$, which means we want all the points where 'y' is greater than or equal to $-2x + 4$. This is the region above the line. Shade this region.

Graphing $y < \frac{5}{2}x + \frac{1}{2}$

  1. Draw the line: Graph the line $y = \frac{5}{2}x + \frac{1}{2}$. This line has a slope of $\frac{5}{2}$ and a y-intercept of $\frac{1}{2}$. Plot the y-intercept (0, 0.5). Then, use the slope to find another point. Go 2 units to the right and 5 units up to find the point (2, 5.5).
  2. Solid or dashed line? Since the inequality is $y < \frac{5}{2}x + \frac{1}{2}$, it does not include the 'equal to' part. This means the line is dashed. A dashed line indicates that the points on the line are not part of the solution.
  3. Shade the region: The inequality is $y < \frac{5}{2}x + \frac{1}{2}$, which means we want all the points where 'y' is less than $\frac{5}{2}x + \frac{1}{2}$. This is the region below the line. Shade this region.

Step 3: Identifying the Solution Region

The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This overlapping region represents all the points (x, y) that satisfy both inequalities simultaneously.

  • Look for the Overlap: Find the area on your graph where both shaded regions coincide. This is your solution region.
  • Boundary Lines: Remember that solid lines are included in the solution, while dashed lines are not. So, points on a solid boundary line are part of the solution, but points on a dashed boundary line are not.

Step 4: Testing a Point (Optional but Recommended)

To be absolutely sure you've identified the correct region, you can test a point from the overlapping region in both original inequalities. If the point satisfies both inequalities, you've likely found the correct solution region.

  • Choose a Point: Pick a point within the overlapping region that's easy to work with. For example, (2, 2) might be a good choice.
  • Test the Point: Plug the point into both original inequalities:
    • For $2x + y \geqslant 4$: $2(2) + 2 \geqslant 4 \Rightarrow 6 \geqslant 4$ (True)
    • For $-5x + 2y < 1$: $-5(2) + 2(2) < 1 \Rightarrow -6 < 1$ (True)

Since (2, 2) satisfies both inequalities, it confirms that we've identified the correct solution region.

Common Mistakes to Avoid

  • Forgetting to Flip the Inequality Sign: Remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
  • Using the Wrong Type of Line: Always check whether the line should be solid (if the inequality includes 'equal to') or dashed (if it doesn't).
  • Shading the Wrong Region: Double-check whether you need to shade above or below the line, depending on the inequality sign.

Example Problems

Let's tackle a couple more examples to solidify your understanding.

Example 1

Solve the system of inequalities:

x+y<3x−y⩾1\begin{aligned} x + y < 3 \\ x - y \geqslant 1 \end{aligned}

  1. Isolate 'y': $y < -x + 3$ and $y \leqslant x - 1$
  2. Graph the Lines: Graph $y = -x + 3$ (dashed) and $y = x - 1$ (solid).
  3. Shade the Regions: Shade below $y = -x + 3$ and below $y = x - 1$.
  4. Identify the Overlap: The overlapping region is the solution.

Example 2

Solve the system of inequalities:

2x−y⩽2x+y>4\begin{aligned} 2x - y \leqslant 2 \\ x + y > 4 \end{aligned}

  1. Isolate 'y': $y \geqslant 2x - 2$ and $y > -x + 4$
  2. Graph the Lines: Graph $y = 2x - 2$ (solid) and $y = -x + 4$ (dashed).
  3. Shade the Regions: Shade above $y = 2x - 2$ and above $y = -x + 4$.
  4. Identify the Overlap: The overlapping region is the solution.

Real-World Applications

Systems of inequalities aren't just abstract math problems. They have real-world applications in various fields.

  • Business: Companies use them to optimize production costs, resource allocation, and pricing strategies.
  • Nutrition: Dieticians use them to create meal plans that meet specific nutritional requirements within certain calorie limits.
  • Engineering: Engineers use them to design structures that can withstand certain loads and stresses.

Conclusion

Solving systems of inequalities might seem tricky at first, but with a little practice, you'll become a pro! Just remember to isolate 'y', graph the lines, shade the correct regions, and identify the overlap. And don't forget to double-check your solution by testing a point.

Keep practicing, and you'll master this skill in no time. Happy solving!