Solving Inequalities Representing Solutions On A Number Line

In this article, we will tackle the mathematical inequality $\frac{1}{2}(20x + 6) \geq x + 30$. Our main goal is to find the solution set for this inequality and correctly represent it on a number line. This involves simplifying the inequality, isolating the variable x, and understanding how to depict the solution graphically. Whether you're a student grappling with algebra or just looking to brush up on your math skills, this guide will provide a step-by-step explanation to ensure you grasp the concepts thoroughly. By the end of this article, you'll be able to confidently solve similar inequalities and interpret their solutions on a number line.

Understanding the Basics of Inequalities

Before we dive into the specifics of our problem, let’s ensure we have a solid grasp of what inequalities are and how they work. Inequalities are mathematical expressions that compare two values, showing that one value is greater than, less than, greater than or equal to, or less than or equal to another value. The symbols we use to represent these relationships are: > (greater than), < (less than), \geq (greater than or equal to), and \leq (less than or equal to). Unlike equations, which have a single solution or a finite set of solutions, inequalities often have a range of values that satisfy the expression. This range is what we call the solution set.

When solving inequalities, we follow similar steps to solving equations, but there’s one crucial difference: when we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the order of the numbers on the number line. For example, if 2 < 4, multiplying both sides by -1 gives -2 > -4. Keeping this rule in mind is essential to correctly solving inequalities.

Another important concept is how to represent the solution set of an inequality graphically on a number line. A number line is a visual representation of all real numbers, typically depicted as a horizontal line with numbers marked at regular intervals. To represent an inequality on a number line, we use either open circles or closed circles (also known as dots) and arrows. An open circle indicates that the endpoint is not included in the solution set (used with > and <), while a closed circle indicates that the endpoint is included (used with \geq and \leq). An arrow extending to the left or right indicates that the solution set includes all numbers less than or greater than the endpoint, respectively. Understanding these basic principles is fundamental to accurately interpreting and representing solutions to inequalities.

Step-by-Step Solution of the Inequality

Now, let's get our hands dirty with the actual inequality: $\frac{1}{2}(20x + 6) \geq x + 30$. To solve this, we'll break it down into manageable steps, ensuring clarity at each stage. First, we need to simplify the expression by distributing the $\frac{1}{2}$ across the terms inside the parentheses. This gives us: $10x + 3 \geq x + 30$.

The next step is to isolate the variable x on one side of the inequality. We can do this by subtracting x from both sides: $10x - x + 3 \geq x - x + 30$, which simplifies to $9x + 3 \geq 30$. Now, we need to get the constant term (3) to the other side. We do this by subtracting 3 from both sides: $9x + 3 - 3 \geq 30 - 3$, which simplifies to $9x \geq 27$.

Finally, to solve for x, we divide both sides of the inequality by 9. Since 9 is a positive number, we don't need to reverse the inequality sign. So, we have: $\frac{9x}{9} \geq \frac{27}{9}$, which simplifies to $x \geq 3$. This is our solution set: all values of x that are greater than or equal to 3. Understanding this algebraic manipulation is crucial for correctly determining the range of values that satisfy the given inequality. Each step, from distributing the initial fraction to the final division, plays a vital role in arriving at the accurate solution.

Representing the Solution on a Number Line

With the solution set determined as $x \geq 3$, the next crucial step is to represent this graphically on a number line. A number line provides a visual representation of all real numbers and is an invaluable tool for understanding inequalities. To depict $x \geq 3$, we first need to locate the number 3 on the number line. Since our inequality includes “equal to” ($\geq$), we use a closed circle (or a filled-in dot) at the number 3. This indicates that 3 itself is part of the solution set.

Next, we need to represent all the numbers that are greater than 3. On a number line, numbers increase as we move from left to right. Therefore, to represent all values greater than 3, we draw an arrow extending from the closed circle at 3 towards the right side of the number line. This arrow signifies that every number to the right of 3, including 3, is a solution to the inequality. If the inequality were $x > 3$, we would use an open circle at 3 to indicate that 3 is not included in the solution set, but the arrow would still extend to the right.

The representation of an inequality on a number line provides an immediate visual understanding of the solution set. It clearly shows the range of values that satisfy the inequality, making it easier to grasp the concept. For $x \geq 3$, the number line visually conveys that 3 and all numbers greater than 3 are valid solutions. This graphical representation is a powerful way to reinforce the algebraic solution and enhance comprehension.

Common Mistakes to Avoid When Solving Inequalities

Solving inequalities can sometimes be tricky, and there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. As we discussed earlier, this is a critical rule, and overlooking it will lead to an incorrect solution set. For instance, if you have the inequality $-2x < 6$, dividing both sides by -2 should give you $x > -3$, not $x < -3$.

Another common mistake is misinterpreting the symbols used in inequalities. Remember that > means