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

by ADMIN 64 views

Hey guys! Today, we're diving deep into the world of inequalities and how to solve them. It might sound a bit intimidating, but trust me, it's like pie once you get the hang of it. We'll break down each problem step-by-step and express our answers in interval notation – a fancy way of showing the range of possible solutions. Let's get started!

Understanding Inequalities

Before we jump into solving, let's quickly recap what inequalities are. Unlike equations that have one specific solution, inequalities show a range of values that satisfy a condition. Think of it like this: instead of saying x equals 5, we might say x is greater than 5 or x is less than or equal to 10.

The main symbols we'll be using are:

  • > Greater than
  • < Less than
  • β‰₯ Greater than or equal to
  • ≀ Less than or equal to

Why is Understanding Inequalities Important? Inequalities form the bedrock of various mathematical and real-world applications. From determining acceptable ranges in engineering to defining constraints in optimization problems, mastering inequalities opens doors to advanced problem-solving. Moreover, inequalities play a crucial role in economics, where they help model supply and demand, and in computer science, where they are used in algorithm design and analysis. Understanding inequalities is not just about manipulating symbols; it's about interpreting relationships between quantities and making informed decisions based on those relationships. By grasping the nuances of inequality symbols and the principles of solving inequalities, you gain a powerful toolset for tackling complex scenarios and developing critical thinking skills. So, let's embark on this journey to unravel the mysteries of inequalities and unlock their potential to solve a myriad of problems.

Interval notation is how we express these ranges. It uses brackets and parentheses to show whether the endpoints are included or excluded. For example:

  • (a, b) means all numbers between a and b, but not including a and b.
  • [a, b] means all numbers between a and b, including a and b.
  • (a, ∞) means all numbers greater than a.
  • (-∞, b] means all numbers less than or equal to b.

Problem 4.1: x + 5 β‰₯ 8

Okay, let's tackle our first inequality: x + 5 β‰₯ 8. Our goal is to isolate x on one side of the inequality. To do this, we need to get rid of that +5. Remember, whatever we do to one side, we gotta do to the other!

  1. Subtract 5 from both sides:
    x + 5 - 5 β‰₯ 8 - 5
  2. Simplify: x β‰₯ 3

Boom! We've got our solution: x is greater than or equal to 3. Now, let's express this in interval notation. Since x can be 3 (or greater), we use a square bracket on the left and the infinity symbol on the right.

Solution in interval notation: [3, ∞)

The solution set for this inequality is all real numbers greater than or equal to 3. To solve this, we use the basic principles of inequality manipulation, which are very similar to those used in solving equations. The key difference is that when we multiply or divide both sides by a negative number, we need to flip the inequality sign. In this case, however, we only needed to subtract a constant from both sides, so the sign remains unchanged. The interval notation [3, ∞) represents all real numbers starting from 3 (inclusive) and extending to positive infinity. This notation clearly and concisely conveys the range of solutions for the inequality, making it an essential tool in mathematical communication and problem-solving. Mastering interval notation allows for precise expression of solution sets and facilitates deeper understanding of the behavior of inequalities in various contexts.

Problem 4.2: d - 12 < 16

Next up, we have d - 12 < 16. Again, we want to isolate d. This time, we're dealing with a subtraction, so we'll use addition to undo it.

  1. Add 12 to both sides: d - 12 + 12 < 16 + 12
  2. Simplify: d < 28

So, d is less than 28. In interval notation, this means we go from negative infinity up to 28, but not including 28. We use a parenthesis to show that 28 is not included.

Solution in interval notation: (-∞, 28)

To solve the inequality d - 12 < 16, we isolate the variable d by adding 12 to both sides of the inequality. This results in d < 28. The interval notation (-∞, 28) represents all real numbers less than 28. The parenthesis on the 28 indicates that 28 is not included in the solution set, meaning that the solution includes all numbers approaching 28 from the left on the number line but does not include 28 itself. This type of notation is crucial for expressing infinite sets of numbers in a concise and unambiguous way. Understanding open intervals, such as (-∞, 28), is fundamental in mathematics as it allows us to precisely define ranges of values that satisfy certain conditions. Mastery of interval notation enhances your ability to communicate mathematical solutions effectively and accurately.

Problem 4.3: 1 - j ≀ -2

Now let's tackle 1 - j ≀ -2. This one has a negative variable, so we need to be a little careful.

  1. Subtract 1 from both sides: 1 - j - 1 ≀ -2 - 1
  2. Simplify: -j ≀ -3

Here's where it gets interesting. We want j, not -j. To get rid of the negative sign, we can multiply (or divide) both sides by -1. But remember the golden rule: when we multiply or divide an inequality by a negative number, we must flip the inequality sign!

  1. Multiply both sides by -1 (and flip the sign): (-1) * -j β‰₯ (-1) * -3
  2. Simplify: j β‰₯ 3

So, j is greater than or equal to 3. Sound familiar? It's the same solution as problem 4.1! In interval notation:

Solution in interval notation: [3, ∞)

In solving the inequality 1 - j ≀ -2, we encounter a crucial step: multiplying or dividing both sides by a negative number. This operation requires flipping the inequality sign to maintain the truth of the statement. After subtracting 1 from both sides, we get -j ≀ -3. To isolate j, we multiply both sides by -1, which changes the inequality sign from ≀ to β‰₯, resulting in j β‰₯ 3. This is a fundamental rule in inequality manipulation, and understanding it is essential for accurately solving inequalities. Flipping the inequality sign when multiplying or dividing by a negative number is a concept that often trips up students, but with practice, it becomes second nature. The solution, j β‰₯ 3, expressed in interval notation as [3, ∞), indicates all real numbers greater than or equal to 3, consistent with our previous solution.

Problem 4.4: 3b + 4 < -5b + 9

Alright, let's move on to 3b + 4 < -5b + 9. This one has b on both sides, so we'll need to gather the b terms together.

  1. Add 5b to both sides: 3b + 4 + 5b < -5b + 9 + 5b
  2. Simplify: 8b + 4 < 9
  3. Subtract 4 from both sides: 8b + 4 - 4 < 9 - 4
  4. Simplify: 8b < 5
  5. Divide both sides by 8: (8b) / 8 < 5 / 8
  6. Simplify: b < 5/8

So, b is less than 5/8. In interval notation:

Solution in interval notation: (-∞, 5/8)

To solve the inequality 3b + 4 < -5b + 9, our strategy involves consolidating the terms with the variable b on one side of the inequality. By adding 5b to both sides, we eliminate the -5b term on the right side, simplifying the inequality to 8b + 4 < 9. Next, we isolate the term with b by subtracting 4 from both sides, resulting in 8b < 5. Finally, we divide both sides by 8 to solve for b, yielding b < 5/8. This step-by-step approach demonstrates the importance of methodical manipulation in solving inequalities. The solution, expressed in interval notation as (-∞, 5/8), represents all real numbers less than 5/8. Systematic problem-solving is a key skill in mathematics, and this example illustrates how breaking down a complex inequality into simpler steps can lead to a clear and accurate solution. Understanding each step and the underlying principles is crucial for mastering inequality manipulation.

Problem 4.5: 3 ≀ -g + 5

Let's take on 3 ≀ -g + 5. This one's similar to problem 4.3, with the variable being negative.

  1. Subtract 5 from both sides: 3 - 5 ≀ -g + 5 - 5
  2. Simplify: -2 ≀ -g
  3. Multiply both sides by -1 (and flip the sign): (-1) * -2 β‰₯ (-1) * -g
  4. Simplify: 2 β‰₯ g

We can also write this as g ≀ 2. So, g is less than or equal to 2. In interval notation:

Solution in interval notation: (-∞, 2]

The inequality 3 ≀ -g + 5 requires careful manipulation to solve for the variable g. Subtracting 5 from both sides gives -2 ≀ -g. To isolate g, we multiply both sides by -1, which necessitates flipping the inequality sign, resulting in 2 β‰₯ g. This can also be expressed as g ≀ 2, emphasizing that g is less than or equal to 2. This step highlights the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. Understanding the direction of inequalities is vital for accurate problem-solving. The solution, expressed in interval notation as (-∞, 2], represents all real numbers less than or equal to 2. This notation clearly communicates the inclusion of 2 in the solution set, thanks to the square bracket on the 2.

Problem 4.6: 12f - 4 ≀ 8f - 7

Here we have 12f - 4 ≀ 8f - 7. Let's gather the f terms on one side and the constants on the other.

  1. Subtract 8f from both sides: 12f - 4 - 8f ≀ 8f - 7 - 8f
  2. Simplify: 4f - 4 ≀ -7
  3. Add 4 to both sides: 4f - 4 + 4 ≀ -7 + 4
  4. Simplify: 4f ≀ -3
  5. Divide both sides by 4: (4f) / 4 ≀ -3 / 4
  6. Simplify: f ≀ -3/4

So, f is less than or equal to -3/4. In interval notation:

Solution in interval notation: (-∞, -3/4]

Solving the inequality 12f - 4 ≀ 8f - 7 involves several steps to isolate the variable f. First, subtract 8f from both sides to get 4f - 4 ≀ -7. Next, add 4 to both sides to isolate the term with f, resulting in 4f ≀ -3. Finally, divide both sides by 4 to solve for f, which gives f ≀ -3/4. This systematic approach of moving terms and isolating the variable is a fundamental technique in solving inequalities. Mastering algebraic manipulation is crucial for efficiently solving such problems. The solution, expressed in interval notation as (-∞, -3/4], represents all real numbers less than or equal to -3/4. The inclusion of -3/4 in the solution set is indicated by the square bracket, demonstrating the precision of interval notation in conveying solution ranges.

Problem 4.7: -12 + 3a ≀ -6a

Let's solve -12 + 3a ≀ -6a. Time to get those a terms together!

  1. Add 6a to both sides: -12 + 3a + 6a ≀ -6a + 6a
  2. Simplify: -12 + 9a ≀ 0
  3. Add 12 to both sides: -12 + 9a + 12 ≀ 0 + 12
  4. Simplify: 9a ≀ 12
  5. Divide both sides by 9: (9a) / 9 ≀ 12 / 9
  6. Simplify (and reduce the fraction): a ≀ 4/3

So, a is less than or equal to 4/3. In interval notation:

Solution in interval notation: (-∞, 4/3]

To solve the inequality -12 + 3a ≀ -6a, the initial step involves consolidating the terms with the variable a. Adding 6a to both sides results in -12 + 9a ≀ 0. Next, we isolate the term with a by adding 12 to both sides, yielding 9a ≀ 12. Finally, dividing both sides by 9 solves for a, giving a ≀ 4/3. Simplifying fractions is a crucial skill in mathematics, and in this case, 12/9 is reduced to 4/3. This step-by-step method illustrates the importance of precision and simplification in solving inequalities. The solution, expressed in interval notation as (-∞, 4/3], represents all real numbers less than or equal to 4/3. The square bracket indicates that 4/3 is included in the solution set, ensuring a comprehensive understanding of the solution range.

Problem 4.8: 4c - 12 > 5c - 12

Let's dive into 4c - 12 > 5c - 12. Another one with variables on both sides!

  1. Subtract 4c from both sides: 4c - 12 - 4c > 5c - 12 - 4c
  2. Simplify: -12 > c - 12
  3. Add 12 to both sides: -12 + 12 > c - 12 + 12
  4. Simplify: 0 > c

We can also write this as c < 0. So, c is less than 0. In interval notation:

Solution in interval notation: (-∞, 0)

Solving the inequality 4c - 12 > 5c - 12 requires the same systematic approach of isolating the variable c. By subtracting 4c from both sides, we simplify the inequality to -12 > c - 12. Adding 12 to both sides further simplifies the inequality to 0 > c, which can also be written as c < 0. This result means that any value of c that is less than 0 will satisfy the original inequality. Understanding the implications of each step is vital in the problem-solving process. The solution, expressed in interval notation as (-∞, 0), represents all real numbers less than 0. The parenthesis on 0 indicates that 0 is not included in the solution set, emphasizing that the solution includes all negative numbers but not zero itself.

Problem 4.9: 6x...

Oops! It looks like the problem is incomplete. We need the full inequality to solve it. But, the process would be the same as the other examples. You know the drill: gather the x terms, isolate x, and express the solution in interval notation.

Key Takeaways

Wow, we've solved a bunch of inequalities today! Here are the key things to remember:

  • Isolate the variable by using inverse operations (addition/subtraction, multiplication/division).
  • When multiplying or dividing by a negative number, flip the inequality sign!
  • Express your answer in interval notation to show the range of solutions.

Inequalities are a fundamental concept in mathematics, and mastering them will open doors to more advanced topics. Keep practicing, and you'll become an inequality-solving pro in no time!

Remember guys, math is not a spectator sport. Get in there, try these problems yourself, and don't be afraid to make mistakes. That's how we learn! Keep up the great work, and I'll catch you in the next math adventure! Continuous practice is the key to mastering any mathematical concept, and inequalities are no exception. So keep practicing, and you'll become an inequality-solving pro in no time!