Solve The Compound Inequality 3x-4 5 Or 1-2x 7

Solving Compound Inequalities: A Comprehensive Guide to 3x - 4 > 5 or 1 - 2x ≥ 7

Compound inequalities involve two or more inequalities joined by the words "and" or "or." Solving them requires understanding the individual inequalities and how the connecting word impacts the solution set. This article will delve into the process of solving the compound inequality 3x - 4 > 5 or 1 - 2x ≥ 7, providing a detailed explanation with examples and tips to help you master this concept.

Understanding Compound Inequalities

Before tackling the specific problem, let's review the fundamentals of compound inequalities. A compound inequality combines two or more inequalities. The solution to a compound inequality depends on the connecting word:

• "or": The solution includes values that satisfy either inequality. This means the solution set is the union of the solution sets of the individual inequalities.

• "and": The solution includes values that satisfy both inequalities simultaneously. The solution set is the intersection of the solution sets of the individual inequalities.

Our problem, 3x - 4 > 5 or 1 - 2x ≥ 7, uses the "or" connector. This means we'll solve each inequality separately and then combine their solution sets.

Solving the First Inequality: 3x - 4 > 5

Let's start by solving the first inequality: 3x - 4 > 5. Our goal is to isolate 'x' on one side of the inequality sign.

1. Add 4 to both sides: This step eliminates the '-4' from the left side. 3x - 4 + 4 > 5 + 4 3x > 9

2. Divide both sides by 3: This isolates 'x'. Since we're dividing by a positive number, the inequality sign remains unchanged. 3x / 3 > 9 / 3 x > 3

This means the solution to the first inequality is all values of 'x' that are greater than 3. We can represent this on a number line as an open circle at 3 and an arrow pointing to the right.

Solving the Second Inequality: 1 - 2x ≥ 7

Now let's tackle the second inequality: 1 - 2x ≥ 7. Again, our aim is to isolate 'x'.

1. Subtract 1 from both sides: This removes the '1' from the left side. 1 - 2x - 1 ≥ 7 - 1 -2x ≥ 6

2. Divide both sides by -2: This is a crucial step. When dividing or multiplying an inequality by a negative number, you must reverse the inequality sign. -2x / -2 ≤ 6 / -2 x ≤ -3

The solution to the second inequality is all values of 'x' that are less than or equal to -3. On a number line, this is represented by a closed circle at -3 (because of the "equal to" part) and an arrow pointing to the left.

Combining the Solutions: The "or" Condition

Since our compound inequality uses "or," we need to find the union of the solution sets of both inequalities. This means we include all values that satisfy either x > 3 or x ≤ -3.

The solution set is: x > 3 or x ≤ -3

This can be expressed in interval notation as: (-∞, -3] ∪ (3, ∞)

The symbol '∪' represents the union, combining both intervals. The parentheses '(' and ')' indicate that the endpoints 3 and -3 are not included (open circle on the number line), while the square bracket ']' indicates that -3 is included (closed circle on the number line).

Graphical Representation

A number line provides a clear visual representation of the solution set.

     <------------------●--------( )-------->
     -∞                   -3           3          ∞

The closed circle at -3 indicates that -3 is included in the solution set. The open circle at 3 indicates that 3 is not included. The arrows show that the solution extends infinitely in both directions.

Practical Applications and Extensions

Understanding compound inequalities is essential in various fields, including:

• Physics: Analyzing motion, forces, and energy often involves inequalities and ranges of values.

• Engineering: Design specifications frequently involve constraints expressed as compound inequalities.

• Economics: Modeling economic scenarios might involve inequalities representing supply and demand relationships.

• Computer Science: Algorithm design and optimization frequently utilize inequalities for setting conditions and constraints.

Solving More Complex Compound Inequalities

While the example provided was relatively straightforward, more complex compound inequalities might involve additional steps or manipulations:

• Inequalities with fractions: Start by clearing the fractions by multiplying both sides by the least common denominator (LCD).

• Inequalities with absolute values: Remember to consider both the positive and negative cases when dealing with absolute value inequalities.

• Multi-step inequalities: Follow the order of operations (PEMDAS/BODMAS) carefully, performing additions/subtractions before multiplications/divisions. Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Tips for Success

• Check your work: Always substitute a value from your solution set back into the original inequality to verify it satisfies the condition.

• Use a number line: Visualizing the solution set on a number line helps you understand the solution's range.

• Practice regularly: The best way to master solving compound inequalities is through consistent practice. Work through various examples, gradually increasing the complexity of the problems.

• Seek help when needed: Don't hesitate to ask for assistance from your teacher, tutor, or classmates if you're struggling with any particular aspect. Online resources and video tutorials can also be beneficial.

This comprehensive guide provides a solid foundation for understanding and solving compound inequalities, especially those involving the "or" condition. By following the steps outlined and practicing regularly, you'll develop the skills necessary to confidently tackle even the most challenging compound inequality problems. Remember that consistent practice and attention to detail are key to mastering this important mathematical concept. Keep practicing, and you'll become proficient in solving these types of problems.