Solve The Inequality 2 4x 3 3 3x 5x

Solving the Inequality: 2 ≤ 4x - 3 ≤ 3 - 3x + 5x

This article provides a comprehensive guide on how to solve the compound inequality 2 ≤ 4x - 3 ≤ 3 - 3x + 5x, explaining each step in detail and offering valuable insights into handling such mathematical problems. We'll explore various methods and emphasize the importance of understanding the underlying principles. This isn't just about finding the solution; it's about mastering the process.

Understanding Compound Inequalities

Before diving into the solution, let's clarify what a compound inequality is. A compound inequality involves two or more inequalities linked together. This particular example is a three-part inequality, meaning it has three parts: a left-hand side, a middle expression, and a right-hand side. The solution will be a range of values for 'x' that satisfy all three parts simultaneously.

The inequality 2 ≤ 4x - 3 ≤ 3 - 3x + 5x can be broken down into two separate inequalities:

• 2 ≤ 4x - 3
• 4x - 3 ≤ 3 - 3x + 5x

However, it's more efficient and elegant to solve it as a single compound inequality.

Step-by-Step Solution

To solve 2 ≤ 4x - 3 ≤ 3 - 3x + 5x, we follow these steps:

Step 1: Simplify the Middle Expression

First, simplify the expression in the middle:

3 - 3x + 5x = 3 + 2x

Now our inequality becomes:

2 ≤ 4x - 3 ≤ 3 + 2x

Step 2: Isolate 'x' in the Left Inequality (2 ≤ 4x - 3)

To isolate 'x', we perform the following operations:

• Add 3 to all parts of the inequality: 2 + 3 ≤ 4x - 3 + 3 ≤ 3 + 2x + 3 This simplifies to 5 ≤ 4x ≤ 5 + 2x

• Subtract 2x from all parts of the inequality: 5 - 2x ≤ 4x - 2x ≤ 5 + 2x - 2x This simplifies to 5 - 2x ≤ 2x ≤ 5

• Solve for x in the left inequality (5 - 2x ≤ 2x): Add 2x to both sides: 5 ≤ 4x. Then divide by 4: 5/4 ≤ x

Step 3: Isolate 'x' in the Right Inequality (2x ≤ 5)

• Divide by 2: 2x/2 ≤ 5/2, simplifying to x ≤ 5/2

Step 4: Combine the Inequalities

We now have two inequalities:

• x ≥ 5/4
• x ≤ 5/2

Combining these, we get the solution:

5/4 ≤ x ≤ 5/2

This means 'x' can be any value greater than or equal to 5/4 (1.25) and less than or equal to 5/2 (2.5).

Verification and Interpretation of the Solution

To verify the solution, let's test some values within the range and outside the range:

• Test x = 1.5: Substitute 1.5 into the original inequality: 2 ≤ 4(1.5) - 3 ≤ 3 - 3(1.5) + 5(1.5). This simplifies to 2 ≤ 3 ≤ 4.5, which is true.

• Test x = 1: Substitute 1 into the original inequality: 2 ≤ 4(1) - 3 ≤ 3 - 3(1) + 5(1). This simplifies to 2 ≤ 1 ≤ 5, which is false because 2 is not less than or equal to 1.

• Test x = 3: Substitute 3 into the original inequality: 2 ≤ 4(3) - 3 ≤ 3 - 3(3) + 5(3). This simplifies to 2 ≤ 9 ≤ 9, which is true, but 3 is outside the derived range.

The verification confirms that the solution 5/4 ≤ x ≤ 5/2 is correct. Any value of 'x' within this interval will satisfy the original compound inequality. Values outside this interval will not.

Alternative Methods of Solving Compound Inequalities

While the step-by-step method is straightforward, other approaches can be used:

• Graphical Method: You can graph the two inequalities on a number line. The solution is the region where the two inequalities overlap.

• Interval Notation: The solution can be expressed using interval notation as [5/4, 5/2]. The square brackets indicate that the endpoints are included in the solution.

Common Mistakes to Avoid

• Incorrect Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions.

• Errors in Adding/Subtracting: Double-check your arithmetic to avoid simple mistakes that can lead to incorrect solutions.

• Incorrect Inequality Signs: Pay close attention to the direction of the inequality signs. Reversing an inequality sign can dramatically alter the solution.

• Forgetting to apply operations to all parts: When adding, subtracting, multiplying, or dividing, make sure you apply the operation to all three parts of the compound inequality.

Expanding the Understanding: Application to Real-World Scenarios

Compound inequalities are not just abstract mathematical concepts; they have practical applications in various real-world situations. For example:

• Budgeting: Suppose you have a budget of $50 to spend on groceries. The cost of milk is $4, and the cost of bread is $3. Let 'x' represent the number of other grocery items you can buy at an average cost of $5 each. You can model the situation using a compound inequality to find the possible number of additional grocery items you can afford.

• Temperature Ranges: Meteorological data often uses inequalities to represent temperature ranges. For example, the temperature might be expressed as being between 15°C and 25°C.

• Speed Limits: Speed limits on highways are defined as a range of permissible speeds.

Mastering the skill of solving compound inequalities is crucial for success in various mathematical and scientific fields. This detailed guide equips you not only with the knowledge to solve this specific inequality but also with the understanding and confidence to tackle similar problems effectively. Remember to practice regularly to reinforce your understanding and refine your problem-solving skills.