Solving Inequalities: Finding The Right Number Line

Hey there, math enthusiasts! Ever found yourself scratching your head over inequalities and number lines? Well, you're in the right place! Today, we're diving deep into the world of inequalities, specifically focusing on how to solve them and visualize the solutions using number lines. We'll be using Edmundo's example, where he tackled the absolute value inequality $|2x - 6| < 4$. Our mission? To understand how to crack this problem and pick out the correct number line that represents the solution. Get ready to flex those math muscles – it's going to be a fun ride!

Understanding Absolute Value Inequalities

First things first, let's get comfy with the basics. What exactly is an absolute value inequality? In simple terms, it's an inequality that involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line. So, whether the number is positive or negative, its absolute value is always non-negative.

For example, $|3| = 3$ and $|-3| = 3$. Both 3 and -3 are 3 units away from zero. When we see an inequality like $|2x - 6| < 4$, it means we're looking for all the values of x that make the expression inside the absolute value (which is 2x - 6) less than 4 units away from zero. Think of it as finding a range on the number line where the distance from 0 is less than 4.

To solve this type of inequality, we need to break it down into two separate inequalities. The absolute value inequality $|2x - 6| < 4$ can be rewritten as: -4 < 2x - 6 < 4. This is because the expression 2x - 6 could be either a positive value less than 4 or a negative value (whose absolute value is less than 4). It is crucial to understand that absolute value inequalities give a range of possible values for the variable. This concept is fundamental to understanding the visual representation on the number line.

In our case, the inequality tells us that the value of the expression 2x - 6 is located within 4 units of zero. That is, it lies between -4 and 4. Breaking down the absolute value into two inequalities gives us a precise range of values.

To grasp the concept thoroughly, consider the general form of an absolute value inequality: |ax + b| < c. This type of inequality has a solution set determined by the two inequalities -c < ax + b and ax + b < c, which can be combined to give the compound inequality -c < ax + b < c. The solution to an absolute value inequality will give us a certain interval of possible values for the given variable. This interval will always be related to the absolute value constraint placed on the given expression.

Solving the Inequality: Step-by-Step

Alright, let's get our hands dirty and solve the inequality $|2x - 6| < 4$. As we discussed, we'll rewrite it as -4 < 2x - 6 < 4. This new form gives us a clear path to find the values of x that satisfy the inequality. The following steps should be followed to find the answer. We will add 6 to each part of the inequality:

-4 + 6 < 2x - 6 + 6 < 4 + 6

This simplifies to:

2 < 2x < 10

Next, we'll divide each part of the inequality by 2:

2/2 < 2x/2 < 10/2

Which simplifies to:

1 < x < 5

This means that x is greater than 1 and less than 5. In other words, the solution set for the inequality is all the numbers between 1 and 5, excluding 1 and 5 themselves. We can represent this solution on a number line.

Now, let's visualize this on a number line. We will need to locate the numbers 1 and 5 on the number line. Since the inequality uses the “less than” symbol (<), we use open circles at 1 and 5. The open circles denote that the values 1 and 5 are not included in the solution set. Then, we shade the region between 1 and 5 to show that all numbers in this interval are part of the solution.

The steps we've followed – rewriting the absolute value inequality, isolating the variable, and simplifying – are essential for solving these types of problems. Remember, the key is to understand what the inequality is asking: find the values of x that make the expression within the absolute value brackets within a certain distance from zero.

Interpreting the Number Line

Okay, we've solved the inequality and know that the solution is 1 < x < 5. But how does this translate to a number line? Well, a number line is a visual representation of all real numbers. It’s like a straight road where each point corresponds to a unique number. In our case, the number line helps us pinpoint the range of values that satisfy our inequality.

When we represent the solution 1 < x < 5 on a number line, we're essentially highlighting all the numbers that fall between 1 and 5. Since our inequality uses the “less than” symbol (<), we use open circles at the numbers 1 and 5. This signifies that 1 and 5 are not included in the solution set. Then, we shade the section of the number line between 1 and 5. This shaded region represents all the numbers that are greater than 1 and less than 5.

Imagine the number line as a highway. The open circles at 1 and 5 are like checkpoints where the solution doesn't include the value at those checkpoints. The shaded region is where all the eligible vehicles (or values of x) can travel freely. It’s a visual way to show the range of values that make the inequality true.

The key to interpreting the number line is to understand the meaning of the symbols used: the open circle (or parenthesis) for