Hey guys! Today, we're diving into the world of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding how to solve inequalities is super important in algebra and beyond. So, let's break it down step by step.

Understanding Inequalities

Before we jump into solving, let's make sure we're all on the same page with what inequalities mean. Unlike equations, which show that two expressions are equal, inequalities show a range of possible values that satisfy a condition. For example:

• x < 5 means x can be any number less than 5 (like 4, 0, -1, etc.).
• x > -2 means x can be any number greater than -2 (like -1, 0, 3, etc.).
• x ≤ 3 means x can be any number less than or equal to 3 (like 3, 1, -2, etc.).
• x ≥ 10 means x can be any number greater than or equal to 10 (like 10, 12, 15, etc.).

Key Symbols: It's crucial to remember what each symbol represents. Getting these mixed up can lead to incorrect solutions.

• < : Less than

• : Greater than

• ≤ : Less than or equal to
• ≥ : Greater than or equal to

Basic Steps for Solving Inequalities

Solving inequalities is a lot like solving equations, but there's one major difference we'll get to in a bit. Here are the basic steps:

1. Simplify: Just like with equations, the first step is to simplify both sides of the inequality as much as possible. This might involve distributing, combining like terms, or clearing fractions.
2. Isolate the Variable: Use inverse operations to isolate the variable on one side of the inequality. This means adding, subtracting, multiplying, or dividing to get the variable by itself.
3. The Flip Rule: Here’s the biggie. If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line.
4. Graph the Solution: Represent the solution on a number line. This gives a visual representation of all the possible values that satisfy the inequality.
5. Write the Solution in Interval Notation: Express the solution using interval notation, which is a concise way to represent a range of numbers.

Let’s dive deeper into each step with examples.

1. Simplify

Simplify both sides of the inequality by distributing and combining like terms. This step makes the inequality easier to work with.

Example:

Solve: 2(x + 3) – 5 < 4x + 1

First, distribute the 2:

2x + 6 – 5 < 4x + 1

Then, combine like terms:

2x + 1 < 4x + 1

2. Isolate the Variable

Use inverse operations to get all terms with the variable on one side and constants on the other. Remember, whatever you do to one side, you must do to the other.

Continuing the Example:

2x + 1 < 4x + 1

Subtract 2x from both sides:

1 < 2x + 1

Subtract 1 from both sides:

0 < 2x

3. The Flip Rule (The Most Important Part!)

This is where inequalities differ significantly from equations. If you multiply or divide by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line.

Example 1 (No Flip Needed):

Let's finish our previous example:

0 < 2x

Divide both sides by 2 (a positive number):

0 < x

Or, written another way:

x > 0

Example 2 (Flip Needed):

Solve: -3x + 6 ≤ 12

Subtract 6 from both sides:

-3x ≤ 6

Divide both sides by -3 (a negative number) – FLIP THE SIGN!

x ≥ -2

4. Graph the Solution

Graphing the solution on a number line provides a visual representation of all possible values that satisfy the inequality. Use a number line. Place a circle (open or closed) at the critical value and shade to the left or right, depending on the inequality sign.

• Open Circle: Used for < and > (the value is not included).
• Closed Circle: Used for ≤ and ≥ (the value is included).

Example 1:

x > 0

• Draw a number line.
• Place an open circle at 0 (since x is greater than, but not equal to, 0).
• Shade to the right (since x is greater than 0).

Example 2:

x ≥ -2

• Draw a number line.
• Place a closed circle at -2 (since x is greater than or equal to -2).
• Shade to the right (since x is greater than -2).

5. Write the Solution in Interval Notation

Interval notation is a concise way to represent a range of numbers. Use parentheses () for values that are not included (corresponding to open circles on the graph) and brackets [] for values that are included (corresponding to closed circles on the graph). Use ∞ (infinity) and -∞ (negative infinity) to represent unbounded intervals.

Example 1:

x > 0

Interval Notation: (0, ∞)

Example 2:

x ≥ -2

Interval Notation: [-2, ∞)

Example 3:

x < 5

Interval Notation: (-∞, 5)

Example 4:

x ≤ 3

Interval Notation: (-∞, 3]

Compound Inequalities

Sometimes, you'll encounter compound inequalities, which involve two or more inequalities joined by "and" or "or."

"And" Inequalities

An "and" inequality requires both inequalities to be true simultaneously. The solution is the intersection of the individual solutions.

Example:

Solve: -3 < 2x + 1 ≤ 5

First, split the compound inequality into two separate inequalities:

-3 < 2x + 1 and 2x + 1 ≤ 5

Solve each inequality:

For -3 < 2x + 1:

-4 < 2x

-2 < x

For 2x + 1 ≤ 5:

2x ≤ 4

x ≤ 2

Combine the solutions:

-2 < x ≤ 2

Graph the solution: A line segment between -2 (open circle) and 2 (closed circle).

Interval Notation: (-2, 2]

"Or" Inequalities

An "or" inequality requires at least one of the inequalities to be true. The solution is the union of the individual solutions.

Example:

Solve: x < -1 or x ≥ 3

Each inequality is already solved. Simply combine the solutions.

Graph the solution: Shade to the left of -1 (open circle) and to the right of 3 (closed circle).

Interval Notation: (-∞, -1) ∪ [3, ∞)

Absolute Value Inequalities

Absolute value inequalities involve absolute value expressions. Remember that the absolute value of a number is its distance from zero, so it’s always non-negative.

Solving Absolute Value Inequalities

1. Isolate the Absolute Value Expression: Get the absolute value expression by itself on one side of the inequality.
2. Split into Two Cases: Create two separate inequalities based on the definition of absolute value.
   • For |x| < a, split into -a < x < a.
   • For |x| > a, split into x < -a or x > a.
3. Solve Each Inequality: Solve each of the resulting inequalities.
4. Combine the Solutions: Determine whether to use "and" or "or" based on the original inequality.

Example 1:

Solve: |x – 2| < 3

Split into two inequalities:

-3 < x – 2 < 3

Add 2 to all parts:

-1 < x < 5

Graph the solution: A line segment between -1 (open circle) and 5 (open circle).

Interval Notation: (-1, 5)

Example 2:

Solve: |2x + 1| ≥ 5

Split into two inequalities:

2x + 1 ≤ -5 or 2x + 1 ≥ 5

Solve each inequality:

For 2x + 1 ≤ -5:

2x ≤ -6

x ≤ -3

For 2x + 1 ≥ 5:

2x ≥ 4

x ≥ 2

Combine the solutions:

x ≤ -3 or x ≥ 2

Graph the solution: Shade to the left of -3 (closed circle) and to the right of 2 (closed circle).

Interval Notation: (-∞, -3] ∪ [2, ∞)

Tips and Tricks for Solving Inequalities

• Always Check Your Work: Substitute values from your solution back into the original inequality to make sure they satisfy the condition.
• Pay Attention to the Flip Rule: This is the most common mistake when solving inequalities. Remember to flip the sign when multiplying or dividing by a negative number.
• Simplify Before Solving: Simplifying the inequality before isolating the variable makes the process easier and reduces the chance of errors.
• Graph Your Solutions: Graphing the solution helps visualize the range of possible values and makes it easier to write the solution in interval notation.
• Practice, Practice, Practice: The more you practice solving inequalities, the more comfortable you'll become with the process.

Conclusion

Alright, guys, that's the rundown on solving inequalities! It might seem like a lot at first, but with practice, you'll get the hang of it. Just remember to simplify, isolate the variable, and always be mindful of the flip rule. Whether you're dealing with basic inequalities, compound inequalities, or absolute value inequalities, these steps will guide you to the correct solution. Keep practicing, and you'll be solving inequalities like a pro in no time!