Solving Inequalities 2x + 3 > 0 And 9x - 2 ≤ 16 A Step By Step Guide

Hey everyone! Today, let's dive into the fascinating world of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like '>', '<', '≥', and '≤'. They're super useful in various real-world scenarios, from figuring out budget constraints to optimizing resources. In this article, we're going to tackle a specific problem: solving the inequalities 2x + 3 > 0 and 9x - 2 ≤ 16. We'll break down each step, making it crystal clear for everyone, whether you're a math whiz or just starting your journey.

Understanding Inequalities

Before we jump into solving our specific inequalities, let's make sure we're all on the same page about what inequalities are and how they work. Think of an equation as a balanced scale – both sides are equal. An inequality, on the other hand, is like a scale that's tilted. One side is greater than, less than, greater than or equal to, or less than or equal to the other side.

• > means 'greater than'
• < means 'less than'
• ≥ means 'greater than or equal to'
• ≤ means 'less than or equal to'

Solving an inequality means finding the range of values for the variable (in our case, 'x') that makes the inequality true. This range is often represented as an interval on a number line.

Key Principles for Solving Inequalities

Solving inequalities is very similar to solving equations, but there's one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line.

Apart from this, we can use the following principles:

1. Addition/Subtraction Property: You can add or subtract the same number from both sides of an inequality without changing the solution.
2. Multiplication/Division Property: You can multiply or divide both sides of an inequality by the same positive number without changing the solution. Remember to flip the sign if you multiply or divide by a negative number.

With these basics covered, we're ready to roll up our sleeves and solve our inequalities.

Solving 2x + 3 > 0

Our first inequality is 2x + 3 > 0. Our goal is to isolate 'x' on one side of the inequality. Here's how we do it:

Step 1: Isolate the Term with 'x'

To get the term with 'x' by itself, we need to get rid of the '+ 3'. We can do this by subtracting 3 from both sides of the inequality. Remember, subtracting the same number from both sides doesn't change the inequality:

2x + 3 - 3 > 0 - 3

This simplifies to:

2x > -3

Step 2: Isolate 'x'

Now, we have 2x > -3. To get 'x' completely by itself, we need to get rid of the '2' that's multiplying it. We can do this by dividing both sides of the inequality by 2. Since 2 is a positive number, we don't need to flip the inequality sign:

2x / 2 > -3 / 2

This simplifies to:

x > -3/2

So, the solution to the inequality 2x + 3 > 0 is x > -3/2. This means any value of 'x' greater than -3/2 will make the inequality true.

Representing the Solution on a Number Line

It's often helpful to visualize the solution on a number line. Draw a number line and mark -3/2 (which is -1.5). Since the inequality is 'x > -3/2' (strictly greater than), we use an open circle at -3/2 to indicate that -3/2 itself is not included in the solution. Then, we shade the region to the right of -3/2, representing all values greater than -3/2.

Solving 9x - 2 ≤ 16

Now, let's tackle our second inequality: 9x - 2 ≤ 16. We'll follow a similar process to isolate 'x'.

Step 1: Isolate the Term with 'x'

To get the term with 'x' by itself, we need to get rid of the '- 2'. We can do this by adding 2 to both sides of the inequality:

9x - 2 + 2 ≤ 16 + 2

This simplifies to:

9x ≤ 18

Step 2: Isolate 'x'

Now, we have 9x ≤ 18. To get 'x' completely by itself, we need to get rid of the '9' that's multiplying it. We can do this by dividing both sides of the inequality by 9. Again, since 9 is a positive number, we don't need to flip the inequality sign:

9x / 9 ≤ 18 / 9

This simplifies to:

x ≤ 2

So, the solution to the inequality 9x - 2 ≤ 16 is x ≤ 2. This means any value of 'x' less than or equal to 2 will make the inequality true.

Representing the Solution on a Number Line

Let's visualize this solution on a number line as well. Draw a number line and mark 2. Since the inequality is 'x ≤ 2' (less than or equal to), we use a closed circle (or a filled-in dot) at 2 to indicate that 2 is included in the solution. Then, we shade the region to the left of 2, representing all values less than 2.

Combining the Solutions: Finding the Intersection

We've solved both inequalities separately. Now, the real magic happens when we combine the solutions. We want to find the values of 'x' that satisfy both inequalities simultaneously. This is called finding the intersection of the solutions.

We know:

• x > -3/2
• x ≤ 2

To find the intersection, we can visualize both solutions on the same number line. We have an open circle at -3/2 shading to the right, and a closed circle at 2 shading to the left.

The intersection is the region where the shaded areas overlap. In this case, it's the region between -3/2 (exclusive) and 2 (inclusive). We can write this as an interval:

(-3/2, 2]

This notation means all values of 'x' greater than -3/2 and less than or equal to 2. The parenthesis '(' indicates that -3/2 is not included, and the bracket ']' indicates that 2 is included.

Real-World Applications of Inequalities

Inequalities aren't just abstract math concepts; they're incredibly useful in real life! Here are a few examples:

• Budgeting: You might have a budget constraint like "I can spend no more than $50 on groceries." This can be written as an inequality.
• Resource Allocation: A factory might need to produce at least 1000 units of a product per day to meet demand. This is another inequality scenario.
• Speed Limits: Speed limits on roads are expressed as inequalities (e.g., the speed must be less than or equal to 65 mph).
• Health and Fitness: You might have a goal like "I want to burn more than 500 calories during my workout." Again, this is an inequality.

Inequalities help us model and solve problems involving constraints, limits, and ranges of values. They're a fundamental tool in mathematics and its applications.

Conclusion: Mastering Inequalities

Great job, everyone! We've successfully solved the inequalities 2x + 3 > 0 and 9x - 2 ≤ 16, and we've even found the intersection of their solutions. Remember, the key to solving inequalities is to isolate the variable while paying close attention to the inequality sign, especially when multiplying or dividing by a negative number.

We've also seen how inequalities are relevant in various real-world scenarios, making them a valuable tool in your mathematical toolkit. Keep practicing, and you'll become a master of inequalities in no time! If you guys have any questions, feel free to ask. Happy problem-solving!