Solving Inequalities A Step-by-Step Guide To Solve For X

Mr. Loba Loba · · 4 min read

Table Of Content

  1. Understanding Compound Inequalities
    1. Step 1: Adding 2 to All Parts
    2. Step 2: Dividing All Parts by 4
  2. Expressing the Solution
    1. Inequality Notation
    2. Interval Notation
    3. Graphing on a Number Line
  3. Common Mistakes to Avoid
    1. Forgetting to Apply Operations to All Parts
    2. Flipping the Inequality Sign Incorrectly
    3. Mixing Up Interval Notation
    4. Not Checking Your Solution
  4. Practice Makes Perfect
    1. Example 1
    2. Example 2
    3. Keep Practicing
  5. Real-World Applications
    1. Budgeting and Finance
    2. Setting Goals
    3. Scientific Research
    4. Engineering
    5. Everyday Decision-Making

Let's dive into solving the compound inequality −6extless=4x−2<14-6 extless= 4x - 2 < 14. This type of problem looks a bit intimidating at first, but don't worry, it's totally manageable. We'll break it down step by step so you can conquer these problems like a pro!

Understanding Compound Inequalities

First things first, what exactly is a compound inequality? Well, it's basically two inequalities joined together. In our case, we have −6extless=4x−2-6 extless= 4x - 2 and 4x−2<144x - 2 < 14. The key to solving these is to treat both inequalities simultaneously. Think of it as a balancing act – whatever you do to one part, you need to do to all parts to keep the inequality in harmony.

Our main goal here is to isolate x in the middle. To do that, we'll perform operations on all three parts of the inequality: the left side (-6), the middle expression (4x - 2), and the right side (14). Remember, we want to get x all by itself, so we'll be using inverse operations to undo the math that's happening to it.

Step 1: Adding 2 to All Parts

The first thing we want to tackle is that “- 2” in the middle. To get rid of it, we'll add 2 to all three parts of the inequality. This keeps everything balanced and moves us closer to isolating x. So, let's add 2:

  • −6+2extless=4x−2+2<14+2-6 + 2 extless= 4x - 2 + 2 < 14 + 2

  • This simplifies to:

  • −4extless=4x<16-4 extless= 4x < 16

See? We've already made progress! The “- 2” is gone from the middle, and we're left with a simpler inequality. We're one step closer to getting x by itself.

Step 2: Dividing All Parts by 4

Now, we have 4x4x in the middle, and we want just x. To get rid of the 4, we'll divide all three parts of the inequality by 4. Remember, when you divide (or multiply) an inequality by a negative number, you need to flip the inequality sign. But in this case, we're dividing by a positive 4, so we don't need to worry about that.

Let's divide:

  • −4/4extless=4x/4<16/4-4 / 4 extless= 4x / 4 < 16 / 4

  • This simplifies to:

  • −1extless=x<4-1 extless= x < 4

And there you have it! We've successfully isolated x. Our solution is −1extless=x<4-1 extless= x < 4. This means that x can be any number greater than or equal to -1, but it must also be less than 4.

Expressing the Solution

Now that we've found the solution, let's talk about how to express it in different ways. There are a few common ways to write the solution to an inequality, and it's good to be familiar with all of them.

Inequality Notation

We've already seen this one – it's the form we ended up with after solving the inequality. In inequality notation, we simply write the inequality as is:

  • −1extless=x<4-1 extless= x < 4

This is a clear and concise way to show the range of values that x can take. It tells us that x is greater than or equal to -1 and less than 4.

Interval Notation

Interval notation is another common way to express the solution. It uses parentheses and brackets to indicate whether the endpoints are included in the solution or not. A bracket “[“ or “]” means the endpoint is included (greater than or equal to, or less than or equal to), while a parenthesis “(“ or “)” means the endpoint is not included (strictly greater than or strictly less than).

For our solution, −1extless=x<4-1 extless= x < 4, we would write it in interval notation as:

  • [−1,4)[-1, 4)

The bracket on the -1 indicates that -1 is included in the solution (because x can be equal to -1), and the parenthesis on the 4 indicates that 4 is not included (because x must be strictly less than 4).

Graphing on a Number Line

Visualizing the solution on a number line can be super helpful. To graph our solution, we'll draw a number line and mark the endpoints, -1 and 4. Then, we'll use a closed circle (or a filled-in dot) on -1 to show that it's included in the solution, and an open circle on 4 to show that it's not included.

Finally, we'll shade the region between -1 and 4 to represent all the values of x that satisfy the inequality. This gives us a visual representation of the solution set.

In summary, graphing the solution −1extless=x<4-1 extless= x < 4 on a number line involves:

  1. Drawing a number line.
  2. Marking -1 with a closed circle (or filled-in dot).
  3. Marking 4 with an open circle.
  4. Shading the region between -1 and 4.

Common Mistakes to Avoid

When solving inequalities, there are a few common pitfalls that students often stumble into. Being aware of these mistakes can help you avoid them and solve inequalities more accurately.

Forgetting to Apply Operations to All Parts

One of the most common mistakes is forgetting to perform an operation on all three parts of the inequality. Remember, whatever you do to one part, you need to do to all parts to maintain the balance. For example, if you add 2 to the middle part, you must also add 2 to the left and right parts.

Flipping the Inequality Sign Incorrectly

As we mentioned earlier, you need to flip the inequality sign when you multiply or divide by a negative number. Forgetting to do this is a frequent error. So, always double-check whether you're multiplying or dividing by a negative, and flip the sign if necessary.

Mixing Up Interval Notation

Interval notation can be tricky at first. Make sure you understand the difference between parentheses and brackets. Parentheses mean the endpoint is not included, while brackets mean it is. Also, always write the smaller number first and the larger number second.

Not Checking Your Solution

It's always a good idea to check your solution by plugging in a value from your solution set into the original inequality. If the inequality holds true, your solution is likely correct. If it doesn't, you've probably made a mistake somewhere along the way.

Practice Makes Perfect

The best way to master solving inequalities is to practice, practice, practice! Work through lots of examples, and don't be afraid to make mistakes – that's how you learn. If you get stuck, review the steps we've covered in this article, and try to identify where you're going wrong.

Example 1

Let's try another example: Solve for x in the inequality 2<3x+5extless=112 < 3x + 5 extless= 11.

  1. Subtract 5 from all parts:

    • 2−5<3x+5−5extless=11−52 - 5 < 3x + 5 - 5 extless= 11 - 5

    • −3<3xextless=6-3 < 3x extless= 6

  2. Divide all parts by 3:

    • −3/3<3x/3extless=6/3-3 / 3 < 3x / 3 extless= 6 / 3

    • −1<xextless=2-1 < x extless= 2

So the solution is −1<xextless=2-1 < x extless= 2. In interval notation, this is (−1,2](-1, 2].

Example 2

Here's another one: Solve for x in the inequality −13<−2x−5<1-13 < -2x - 5 < 1.

  1. Add 5 to all parts:

    • −13+5<−2x−5+5<1+5-13 + 5 < -2x - 5 + 5 < 1 + 5

    • −8<−2x<6-8 < -2x < 6

  2. Divide all parts by -2 (and flip the inequality signs!):

    • −8/−2>−2x/−2>6/−2-8 / -2 > -2x / -2 > 6 / -2

    • 4>x>−34 > x > -3

It's more common to write this with the smaller number on the left, so we can rewrite it as:

  • −3<x<4-3 < x < 4

In interval notation, this is (−3,4)(-3, 4).

Keep Practicing

Try solving various inequalities, both simple and complex. The more you practice, the more comfortable you'll become with the process. Remember to always check your solutions, and don't be afraid to ask for help if you need it.

Solving inequalities is a fundamental skill in algebra, and it's something you'll use in many different areas of math. So, put in the time and effort to master it, and you'll be well on your way to success!

Real-World Applications

Inequalities aren't just abstract math concepts – they actually show up in a lot of real-world situations! Understanding how to solve them can be incredibly useful in various fields and everyday scenarios. Let's take a look at some examples.

Budgeting and Finance

One common application is in budgeting. Suppose you have a certain amount of money to spend, and you want to make sure you don't go over budget. You can use an inequality to represent this situation. For example, if you have $200 to spend on groceries and you've already spent $50, you can write an inequality like this:

$50 + x extless= $200

Here, x represents the amount of money you can still spend. Solving this inequality will tell you the maximum amount you can spend without exceeding your budget.

Setting Goals

Inequalities can also be used to set goals. For instance, if you want to save at least $1000 by the end of the year, you can set up an inequality to track your progress. If you save $100 each month, you can write:

$100m extgreater= $1000

where m is the number of months. Solving this inequality will tell you how many months you need to save to reach your goal.

Scientific Research

In scientific research, inequalities are often used to define ranges and constraints. For example, in a medical study, researchers might want to determine the effective dosage range for a new drug. They might use inequalities to specify the minimum and maximum dosages that are safe and effective.

Engineering

Engineers use inequalities to design structures and systems that meet certain specifications. For example, when designing a bridge, engineers need to ensure that the bridge can support a certain weight limit. They can use inequalities to represent the maximum load the bridge can handle.

Everyday Decision-Making

Even in everyday life, we use inequalities all the time, often without even realizing it. For example, when deciding how much time to spend on a task, we might think,

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