Solving & Graphing Linear Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of linear inequalities. We're going to solve some, represent the solutions graphically on a number line, and then express those solutions using interval notation. This guide is designed to make it super clear and easy to understand, so don't worry if you're feeling a bit lost – we'll break everything down step by step. Understanding inequalities is super important in math, because it pops up in a lot of different areas! So, let's get started!

Understanding Linear Inequalities: The Basics

First things first, what exactly are linear inequalities? Well, they're mathematical statements that compare two expressions using inequality symbols. Instead of an equals sign (=), we use symbols like:

• <: Less than
• >: Greater than
• ≤: Less than or equal to
• ≥: Greater than or equal to

The goal when solving a linear inequality is to find all the values of the variable (usually 'x') that make the inequality true. The process is similar to solving linear equations, with a few key differences that we will discuss later. The solution to an inequality is often an infinite set of numbers, which is why we represent them on a number line and with interval notation. Think of it like this: an equation finds one specific point, while an inequality finds a whole range of points that fit the criteria. Now, understanding how to manipulate inequalities is vital. We want to isolate the variable on one side. Remember to do the same operation on both sides of the inequality to keep it balanced.

Properties of Inequalities

There are a few important properties of inequalities that we need to keep in mind:

1. Addition and Subtraction: You can add or subtract the same number from both sides of an inequality without changing the solution. This is just like equations!
2. Multiplication and Division by a Positive Number: If you multiply or divide both sides of an inequality by a positive number, the inequality sign stays the same. Easy peasy!
3. Multiplication and Division by a Negative Number: This is the crucial part! If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line. For example, -2 is less than -1, but 2 is greater than 1. You got it?

So, before we jump into the examples, let's just recap: isolate the variable, do the same thing to both sides (like equations), and flip the sign when you multiply or divide by a negative number. That's pretty much the core of it!

Let's Solve Some Linear Inequalities!

Alright, let's get our hands dirty and solve some actual inequalities. We'll go through each step carefully, so you can see how it all comes together.

a) Solving $4 - 5x < 19$

Here's our first problem: $4 - 5x < 19$. Let's solve it together:

1. Isolate the x term: First, we want to get the term with 'x' by itself. We can start by subtracting 4 from both sides of the inequality: $4 - 5x - 4 < 19 - 4$ This simplifies to: $-5x < 15$

2. Isolate x: Now, we need to get 'x' completely alone. We do this by dividing both sides by -5. Remember that super important rule? Since we're dividing by a negative number, we need to flip the inequality sign! This is where a lot of people make mistakes, so pay close attention. rac{-5x}{-5} > rac{15}{-5} This simplifies to: $x > -3$

   So, the solution to the inequality is $x > -3$. This means any number greater than -3 will make the original inequality true.

Graphing the Solution on a Number Line

Now, let's graph this solution on a number line. Here's how:

1. Draw a number line: Draw a straight line and mark some numbers on it. Include -3 and a few numbers on either side (like -4, -2, -1, 0, 1, etc.) to give some context.
2. Mark the endpoint: Since our solution is $x > -3$, we want to show that -3 is not included in the solution. We do this by drawing an open circle (or a parenthesis) at -3. An open circle means the point isn't included. If the inequality had been $x extbf{≥} -3$, we would have used a closed circle (or a bracket), because -3 would be included.
3. Shade the solution: The solution is all the numbers greater than -3. So, starting from the open circle at -3, we shade the number line to the right. The shading shows all the values of x that make the inequality true. The number line will have an open circle on -3 and the line will be shaded to the right.

Interval Notation

Finally, let's express the solution in interval notation. Interval notation uses parentheses and brackets to show the range of values in the solution. Remember these guidelines:

• Parentheses (): Used for values that are not included in the solution (like our open circle).
• Brackets []: Used for values that are included in the solution (like a closed circle).
• Infinity ∞ and negative infinity -∞: Always use parentheses because infinity is not a specific number.

For our solution, $x > -3$, the solution goes from -3 (not included) to infinity. Therefore, the interval notation is: (-3, ∞). That's it! We've solved, graphed, and notated our first inequality. Well done!

Tackling More Inequalities

Let's keep the ball rolling and solve another inequality. We will go into more examples now to ensure you understand it.

b) Solving $3(x + 3) extbf{≥} 4x + 5$

Let's get started on $3(x + 3) extbf{≥} 4x + 5$. Here’s how:

1. Distribute: First, we need to simplify the left side by distributing the 3: $3x + 9 extbf{≥} 4x + 5$
2. Get x terms together: Next, let's get all the 'x' terms on one side. Subtract $3x$ from both sides: $3x + 9 - 3x extbf{≥} 4x + 5 - 3x$ This simplifies to: $9 extbf{≥} x + 5$
3. Isolate x: Now, subtract 5 from both sides to isolate 'x': $9 - 5 extbf{≥} x + 5 - 5$ This simplifies to: $4 extbf{≥} x$ This can also be written as $x extbf{≤} 4$. So, the solution is $x extbf{≤} 4$. This time the sign flips to less than or equal to!

Graphing the Solution on a Number Line

1. Draw a number line: Draw a number line and mark some numbers, including 4 and a few on either side.
2. Mark the endpoint: The solution is $x extbf{≤} 4$. This means 4 is included. Therefore, we use a closed circle (or a bracket) at 4.
3. Shade the solution: The solution includes all numbers less than or equal to 4. Therefore, shade the number line to the left of 4.

Interval Notation

Since our solution is $x extbf{≤} 4$, it goes from negative infinity up to and including 4. Thus, the interval notation is: $(-\infty, 4]$. Notice that we use a bracket ] on the 4 to show that it is included.

Working with Compound Inequalities

Now, let's tackle a compound inequality. This is an inequality that combines two inequalities into one statement. They're not as scary as they sound, I promise! The third problem is a compound inequality.

c) Solving $4 extbf{≤} 3x + 1 < 12$

Here’s how we solve $4 extbf{≤} 3x + 1 < 12$:

1. Isolate the x term: The goal is to get 'x' alone in the middle. We'll start by subtracting 1 from all three parts of the inequality: $4 - 1 extbf{≤} 3x + 1 - 1 < 12 - 1$ This simplifies to: $3 extbf{≤} 3x < 11$

2. Isolate x: Now, divide all three parts by 3: rac{3}{3} extbf{≤} rac{3x}{3} < rac{11}{3} This simplifies to: 1 extbf{≤} x < rac{11}{3}

   So, the solution is 1 extbf{≤} x < rac{11}{3}. This means 'x' is greater than or equal to 1, but less than 11/3.

Graphing the Solution on a Number Line

1. Draw a number line: Draw a number line and mark some numbers. Include 1 and 11/3 (which is approximately 3.67), and some numbers in between and on either side.
2. Mark the endpoints: At 1, we use a closed circle (or a bracket) because $x$ is greater than or equal to 1. At 11/3, we use an open circle (or a parenthesis) because $x$ is strictly less than 11/3.
3. Shade the solution: Since $x$ is between 1 and 11/3, we shade the number line between these two points.

Interval Notation

For our solution, 1 extbf{≤} x < rac{11}{3}, the interval notation is: [1, rac{11}{3}). We use a bracket [ on the 1 to include it, and a parenthesis ) on the 11/3 to exclude it.

Tips and Tricks for Solving Inequalities

• Always double-check your work: It's easy to make small mistakes, so always take a moment to review your steps, especially when flipping the inequality sign. Reread the steps. It is important to remember what steps you have done to avoid errors.
• Test a value: After you think you've solved an inequality, pick a number from your solution set and plug it back into the original inequality to see if it works. This is a great way to catch any errors.
• Practice, practice, practice! The more you practice, the more comfortable you'll become with solving inequalities. Try different examples. There are plenty of resources available online and in textbooks. Make your own examples.

Conclusion: You Got This!

Great job, guys! You've learned how to solve linear inequalities, graph their solutions on a number line, and express them in interval notation. Remember the key steps: isolate the variable, pay attention to the sign when multiplying or dividing by a negative number, and practice! Math can be challenging, but with the right approach and effort, you can totally master it. Keep up the great work, and you'll be solving inequalities like a pro in no time! Remember to always believe in yourself and your abilities. You are doing great and you can do it!