Solving Inequalities: A Step-by-Step Guide With Interval Notation

Hey guys! Let's dive into solving inequalities today. Inequalities might seem a bit tricky at first, but don't worry, we'll break it down step by step. In this guide, we'll tackle an example inequality and show you exactly how to solve it and express the solution using interval notation. So, grab your pencils and let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are all about. Unlike equations that have a single solution, inequalities represent a range of possible solutions. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). When we solve an inequality, we're essentially finding all the values that make the inequality true. Expressing these solutions in interval notation is a neat way to represent these ranges.

The Problem: Solving $\frac{1}{5} k-\frac{23}{25}<\frac{17}{25} k+1$

Okay, let's get to the main event. Our mission is to solve the inequality $\frac{1}{5} k-\frac{23}{25}<\frac{17}{25} k+1$ and write the answer in interval notation. This might look a little intimidating with all the fractions, but we'll take it one step at a time. The key is to follow the same principles we use for solving regular equations, with one extra rule to keep in mind – we'll talk about that later.

Step 1: Clearing the Fractions

Fractions can sometimes make things look more complicated than they really are. So, our first move is to get rid of them. To do this, we need to find the least common denominator (LCD) of all the fractions in the inequality. In this case, we have denominators of 5 and 25. The LCD of 5 and 25 is 25. Now, we'll multiply both sides of the inequality by 25. This will clear out the fractions and make our equation much easier to work with.

Multiplying both sides by 25, we get:

$25 * (\frac{1}{5} k-\frac{23}{25}) < 25 * (\frac{17}{25} k+1)$

Distribute the 25 on both sides:

$25 * \frac{1}{5} k - 25 * \frac{23}{25} < 25 * \frac{17}{25} k + 25 * 1$

Simplify each term:

$5k - 23 < 17k + 25$

See how much cleaner that looks? We've successfully cleared the fractions!

Step 2: Isolating the Variable

Now that we have a simpler inequality, our next goal is to isolate the variable k. This means getting all the terms with k on one side of the inequality and all the constant terms on the other side. It doesn't matter which side we choose for k, but it's often easier to move the terms in a way that keeps the coefficient of k positive. In this case, let's move the 5k term to the right side of the inequality.

Subtract 5k from both sides:

$5k - 23 - 5k < 17k + 25 - 5k$

Simplify:

$-23 < 12k + 25$

Next, let's move the constant term 25 to the left side. Subtract 25 from both sides:

$-23 - 25 < 12k + 25 - 25$

Simplify:

$-48 < 12k$

Step 3: Solving for k

We're almost there! Now we just need to get k by itself. To do this, we'll divide both sides of the inequality by the coefficient of k, which is 12.

Divide both sides by 12:

$\frac{-48}{12} < \frac{12k}{12}$

Simplify:

$-4 < k$

Or, we can rewrite this as:

$k > -4$

This tells us that k is greater than -4. Now, remember that extra rule we mentioned earlier? Here it is: If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. In this case, we divided by a positive number (12), so we don't need to worry about flipping the sign. But it's crucial to keep this rule in mind for other problems!

Step 4: Expressing the Solution in Interval Notation

We've solved the inequality and found that k > -4. Now, let's express this solution in interval notation. Interval notation is a way of writing sets of numbers using intervals. We use parentheses ( ) to indicate that an endpoint is not included in the interval and brackets [ ] to indicate that an endpoint is included.

Since k is greater than -4, this means that k can be any number greater than -4, but not -4 itself. On a number line, we would represent this with an open circle at -4 and an arrow extending to the right, indicating all the numbers greater than -4. In interval notation, we write this as:

$(-4, \infty)$

Here's what this notation means:

• (-4: The left endpoint of the interval is -4, but it's not included (hence the parenthesis).
• ∞: The right endpoint is positive infinity, indicating that the interval extends indefinitely to the right. We always use a parenthesis with infinity because infinity is not a number, but a concept.

So, the solution to the inequality $\frac{1}{5} k-\frac{23}{25}<\frac{17}{25} k+1$ in interval notation is $(-4, \infty)$.

Wrapping Up

And that's it! We've successfully solved the inequality and expressed the solution in interval notation. Let's quickly recap the steps we took:

1. Cleared the fractions by multiplying both sides of the inequality by the least common denominator.
2. Isolated the variable by moving terms around.
3. Solved for k by dividing both sides by the coefficient of k.
4. Expressed the solution in interval notation.

Solving inequalities is a fundamental skill in algebra, and mastering it opens the door to more advanced math topics. Remember the key rule about flipping the inequality sign when multiplying or dividing by a negative number, and you'll be well on your way to conquering inequalities like a pro!

Keep practicing, and you'll become a master of inequalities in no time. If you have any questions, feel free to ask in the comments below. Happy solving!