Solving Inequalities: A Comprehensive Guide with Step-by-Step Solutions

In the realm of algebra, inequalities can be a challenging but essential topic. This article provides a detailed guide to solving inequalities using a series of examples and step-by-step methods. By following the solutions to the provided inequalities, you will gain a better understanding of algebraic techniques and how to apply them effectively.

Example 1: Solving -2 ≤ 5 - 3x/4 ≤ 1/2

Step 1: Subtract 5 from all parts of the inequality

To simplify the inequality, we start by subtracting 5 from all parts.

-2 - 5 ≤ 5 - 3x/4 - 5 ≤ 1/2 - 5

This simplifies to:

-7 ≤ -3x/4 ≤ -9/2

Step 2: Multiply all parts of the inequality by -4

In order to eliminate the fraction, we multiply each part by -4, remembering to reverse the inequality signs.

-7 * (-4) ≥ -3x/4 * (-4) ≥ -9/2 * (-4)

This simplifies to:

28 ≥ 3x ≥ 18

Step 3: Divide all parts of the inequality by 3

To solve for x, we divide all parts by 3, keeping the inequality signs the same.

28/3 ≥ x ≥ 18/3

This simplifies to:

28/3 ≥ x ≥ 6

Therefore, the solution to the inequality -2 ≤ 5 - 3x/4 ≤ 1/2 is 6 ≤ x ≤ 28/3.

Example 2: Solving -2 ≤ (5 - 3x)/4 ≤ 1/2

This example is essentially the same as the first example. The main difference is how the inequality is written, but the solution process remains the same.

Step 1: Subtract 5 from all parts of the inequality

-2 - 5 ≤ (5 - 3x)/4 - 5 ≤ 1/2 - 5

This simplifies to:

-7 ≤ (5 - 3x - 20)/4 ≤ -9/2

-7 ≤ -15 - 3x/4 ≤ -9/2

Step 2: Multiply all parts by -4 (reversing the inequality signs)

-7 * (-4) ≥ -15 - 3x/4 * (-4) ≥ -9/2 * (-4)

This simplifies to:

28 ≥ 3x 60 ≥ 18

Step 3: Subtract 60 from all parts of the inequality

28 - 60 ≥ 3x 60 - 60 ≥ 18 - 60

This simplifies to:

-32 ≥ 3x ≥ -42

Step 4: Divide all parts by 3

-32/3 ≥ x ≥ -42/3

This simplifies to:

-32/3 ≥ x ≥ -14

Therefore, the solution to the inequality -2 ≤ (5 - 3x)/4 ≤ 1/2 is -32/3 ≤ x ≤ -14.

Additional Tips for Solving Inequalities

Tips for Solving Inequalities

Understand the Basics: Always remember the rules of inequality, such as reversing the sign when multiplying or dividing by a negative number. Isolate the Variable: Try to isolate the variable you are solving for on one side of the inequality. Check Your Solution: After solving, substitute a solution back into the original inequality to check if it holds true.

Conclusion

Mastering the art of solving inequalities is crucial for success in algebra and many other advanced mathematical topics. By following the step-by-step methods outlined in this guide, you can ensure that you are accurately solving inequalities and gaining a deeper understanding of algebraic principles.

Keywords: inequality, solving inequalities, algebraic solutions