Solve And Graph Linear Inequalities: A Step-by-Step Guide

Understanding Linear Inequalities

Welcome! Today, we're diving into the world of linear inequalities. Think of them as cousins to linear equations, but instead of an equal sign, they use symbols like '<' (less than), '>' (greater than), '

≤' (less than or equal to), or '≥' (greater than or equal to). These inequalities help us describe a range of possible values rather than a single, precise answer. For instance, if you're saving money and want to buy a game that costs less than $60, you'd use an inequality. If you're working with a problem involving a linear expression like $3(2x+1)+4x$ and want to find out when its value is less than 63, you're dealing with a linear inequality.

Why are these important? In mathematics and real-world applications, we often encounter situations where a condition isn't met exactly but within a certain range. For example, a bridge might have a weight limit (greater than or equal to a certain value), or a recipe might require a certain amount of an ingredient but not exceed a maximum amount. Understanding how to solve and graph these inequalities is a fundamental skill that opens doors to solving more complex problems. We'll break down the process step-by-step, making it easy to follow, even if you're new to this concept. Our goal is to not only find the solution set but also to visualize it on a number line, which is where the graphing part comes in.

Mastering linear inequalities will equip you with a powerful tool for analyzing and understanding mathematical relationships. It's about more than just crunching numbers; it's about interpreting conditions and representing them visually. We'll cover everything from simplifying the expression to plotting the final solution, ensuring you feel confident every step of the way. Let's get started on this mathematical journey!

Step 1: Simplify the Inequality

Our journey begins with tackling the inequality: $3(2x+1)+4x < 63$. The first crucial step in solving any inequality, including this one, is to simplify both sides of the inequality as much as possible. This often involves using the distributive property and combining like terms. Think of it as tidying up the expression before you start isolating the variable. In our case, the left side of the inequality has parentheses that need to be dealt with. We'll apply the distributive property by multiplying the 3 outside the parentheses by each term inside: $3 \times 2x$ and $3 \times 1$. This gives us $6x + 3$. Now, our inequality looks like this: $6x + 3 + 4x < 63$.

See that? We've made it a bit cleaner already. But we're not done simplifying yet. We have two terms with 'x' on the left side: $6x$ and $4x$. These are like terms, meaning they have the same variable raised to the same power. We can combine them by adding their coefficients: $6x + 4x = 10x$. So, after combining the like terms, our inequality transforms into $10x + 3 < 63$. This simplified form is much easier to work with as we move towards isolating our variable, 'x'.

Remember, the goal of simplification is to get the inequality into its most basic form, where the variable terms are combined and the constant terms are combined on each side. This makes the subsequent steps of solving for the variable straightforward. Don't shy away from applying the distributive property or combining like terms – these are fundamental algebraic techniques that will serve you well. By carefully performing these operations, we ensure that we maintain the integrity of the inequality and pave the way for a clear and accurate solution. This initial step is the bedrock upon which the rest of the solution is built, so take your time and make sure every calculation is precise.

Step 2: Isolate the Variable

Now that we have our simplified inequality, $10x + 3 < 63$, the next logical step is to isolate the variable 'x'. Our aim is to get 'x' all by itself on one side of the inequality sign. To do this, we'll use inverse operations, just like we would with an equation. The key principle here is that whatever operation you perform on one side of the inequality, you must perform the same operation on the other side to maintain the balance. It's like a seesaw – you have to keep things even!

First, we need to get rid of the constant term on the same side as 'x'. In $10x + 3$, the constant is +3. The inverse operation of adding 3 is subtracting 3. So, we will subtract 3 from both sides of the inequality: $10x + 3 - 3 < 63 - 3$. Performing the subtraction, we get $10x < 60$. Look at that – 'x' is getting closer to being alone!

Next, we need to deal with the coefficient of 'x', which is 10. Currently, 'x' is being multiplied by 10. The inverse operation of multiplication is division. Therefore, we will divide both sides of the inequality by 10: $\frac{10x}{10} < \frac{60}{10}$. This simplifies to $x < 6$.

There's one crucial rule to remember when isolating variables in inequalities: if you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality sign. For example, if we had $-2x < 10$, dividing by -2 would flip the sign to $x > -5$. In our case, we divided by a positive number (10), so the inequality sign '<' remains unchanged.

So, the solution to our inequality is $x < 6$. This means that any number less than 6 will satisfy the original inequality. This is the algebraic solution. But how do we show this visually? That's where graphing comes in, and it's our next exciting step!

Step 3: Graph the Solution on a Number Line

We've successfully solved the inequality and found that $x < 6$. Now, it's time to bring our solution to life by graphing it on a number line. This visual representation makes it incredibly easy to see all the possible values of 'x' that satisfy the inequality. Think of a number line as a ruler that extends infinitely in both directions, marked with integers.

To graph $x < 6$, we first locate the number 6 on our number line. This is our boundary point. Since the inequality is strictly 'less than' (<) and not 'less than or equal to' (≤), the number 6 itself is not included in our solution set. To indicate this, we use an open circle at the point 6 on the number line. An open circle signifies that the boundary point is excluded.

If the inequality had been $x

≤

6$, we would have used a closed circle (or a filled-in circle) at 6 to show that 6 is included in the solution. But for $x < 6$, the open circle is correct.

Now, we need to represent all the numbers that are less than 6. On a standard horizontal number line, numbers decrease as you move to the left and increase as you move to the right. Since we want values less than 6, we will draw a bold arrow or line extending from the open circle at 6 and pointing towards the left, covering all the numbers smaller than 6. This shaded region and the arrow clearly indicate that any number to the left of 6, no matter how small (e.g., 5, 0, -10, -1000), is a valid solution to our inequality.

When you create your graph, make sure to label the important points, like the boundary number (6 in this case), and clearly show the direction of the solution set. You might also want to include a few other integers around the boundary point (like 5 and 7) to give context to the number line. The open circle at 6 and the shading to the left visually communicate the solution $x < 6$ in a way that's immediately understandable. It’s a powerful way to represent infinite sets of numbers.

Conclusion: Mastering Inequalities

We've successfully navigated the process of solving and graphing the linear inequality $3(2x+1)+4x < 63$. We started by simplifying the expression, which led us to the inequality $10x + 3 < 63$. Then, we skillfully isolated the variable 'x' using inverse operations, revealing the solution $x < 6$. Finally, we translated this algebraic solution into a visual representation on a number line, marking the boundary with an open circle and shading to the left to indicate all numbers less than 6 are part of the solution set.

Understanding how to solve and graph linear inequalities is a fundamental skill in mathematics. It allows us to represent conditions and constraints that aren't exact but rather fall within a range. This concept extends far beyond simple algebra, finding applications in areas like optimization problems, resource allocation, and understanding tolerances in engineering. The ability to manipulate these inequalities and visualize their solutions provides a clear and intuitive grasp of mathematical relationships that govern many real-world scenarios.

Remember the key steps: simplify, isolate, and graph. Each step builds upon the last, leading you to a complete understanding of the inequality's solution. Always pay close attention to the inequality symbol (<, >, ≤, ≥) as it dictates whether the boundary point is included (closed circle) or excluded (open circle) and the direction of your graph. Also, be mindful of the rule about reversing the inequality sign when multiplying or dividing by a negative number – a common pitfall to avoid.

Keep practicing with different types of linear inequalities, and you'll find your confidence and proficiency grow. This skill will serve you well in future mathematical endeavors and problem-solving scenarios. For further exploration and practice, you can refer to resources like Khan Academy for comprehensive lessons and exercises on inequalities, or check out the detailed explanations on the Math is Fun website. These platforms offer a wealth of information to deepen your understanding.