Solve Absolute Value Inequalities & Graph On Number Line
Hey guys! Today, we're diving into the exciting world of inequalities, but not just any inequalities – we're tackling those involving absolute values! Absolute values can seem a little tricky at first, but don't worry, we'll break it down step by step. We'll not only solve these inequalities but also represent their solutions beautifully on a number line. So, grab your pencils and let's get started!
Understanding Absolute Value
Before we jump into solving, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero, regardless of direction. This means it's always non-negative. We denote the absolute value of x as |x|. For example, |3| = 3 and |-3| = 3. Both 3 and -3 are three units away from zero.
The key to solving absolute value inequalities lies in understanding how to deal with this dual nature of positive and negative distances. When we have an inequality like |x| ≤ 5, it means we're looking for all numbers whose distance from zero is less than or equal to 5. This includes numbers like 0, 2, -4, and of course, 5 and -5 themselves. Similarly, if we have |x| ≥ 5, we're looking for numbers whose distance from zero is greater than or equal to 5, like 6, -7, or even 100. This understanding is crucial for tackling the problems ahead.
Remember, the absolute value essentially creates two scenarios we need to consider: the positive case and the negative case. This is because a number within the absolute value could have originally been positive or negative and still result in the same absolute value. Keeping this in mind will make solving these inequalities much smoother. It's like having a secret code where you need to decipher both possibilities to unlock the solution. So, let’s keep this in our toolkit as we move forward and solve some exciting problems!
Solving |x| ≤ 5
Alright, let's kick things off with our first inequality: |x| ≤ 5. This inequality asks us to find all values of x whose absolute value is less than or equal to 5. In other words, we need to find all numbers that are within 5 units of zero on the number line.
To solve this, we can break it down into two separate inequalities. Remember, the absolute value means x could be either positive or negative. So, we have two cases to consider:
- x ≤ 5 (the positive case): This simply states that x is less than or equal to 5.
- -x ≤ 5 (the negative case): To solve this, we multiply both sides by -1, which flips the inequality sign, giving us x ≥ -5.
Combining these two inequalities, we get -5 ≤ x ≤ 5. This means x can be any number between -5 and 5, including -5 and 5 themselves. It’s like setting boundaries for x – it can roam freely within this range.
Now, let's represent this solution on a number line. We'll draw a number line and mark -5 and 5. Since x can be equal to -5 and 5, we'll use closed circles (or brackets) at these points to indicate that they are included in the solution. Then, we'll shade the region between -5 and 5, showing that all numbers in this interval are solutions to the inequality.
Visualizing the solution on a number line makes it super clear. It’s like seeing a map of all the possible values x can take. This graphical representation is a powerful tool in understanding inequalities, especially when dealing with absolute values. So, there you have it! We've successfully solved our first absolute value inequality and visualized the solution. Let’s keep this momentum going as we tackle the next one!
Solving |(x+6)/2| ≥ 12
Next up, we have the inequality |(x+6)/2| ≥ 12. This one looks a little more complex, but don't worry, we'll tackle it using the same principles we learned earlier. Remember, the key is to break it down into two separate cases based on the absolute value.
This inequality tells us that the absolute value of the expression (x+6)/2 is greater than or equal to 12. This means the expression inside the absolute value is either greater than or equal to 12, or less than or equal to -12. So, let's set up our two cases:
- (x+6)/2 ≥ 12 (the positive case): To solve this, we first multiply both sides by 2, which gives us x + 6 ≥ 24. Then, we subtract 6 from both sides, resulting in x ≥ 18.
- (x+6)/2 ≤ -12 (the negative case): Again, we start by multiplying both sides by 2, giving us x + 6 ≤ -24. Subtracting 6 from both sides, we get x ≤ -30.
So, our solution is x ≥ 18 or x ≤ -30. Notice the
