Solving the Inequality 2x - 5 ≥ 11 and Related Concepts

Algebra is a fundamental branch of mathematics that helps us solve various types of problems, including inequalities. One such problem is the inequality 2x - 5 ≥ 11. This article will guide you through solving this problem step by step, and also explore the mathematical concepts behind it.

Understanding the Problem

The inequality presented is 2x - 5 ≥ 11. This is a linear inequality, where the variable x is multiplied by a constant, and there is a constant term on the left-hand side, which is subtracted from twice the variable. The symbol ≥ means 'greater than or equal to,' indicating that the solution set includes the boundary value as well as all values greater than it.

Step-by-Step Solution

Step 1: Subtract 5 from Both Sides

The first step in solving the inequality is to isolate the term containing the variable x. We start by subtracting 5 from both sides of the inequality:

2x - 5 - 5 ≥ 11 - 5

This simplifies to:

2x - 10 ≥ 6

Step 2: Divide Both Sides by 2

Next, we divide both sides of the inequality by 2 to solve for x:

(2x - 10) / 2 ≥ 6 / 2

This simplifies to:

2x / 2 - 10 / 2 ≥ 6 / 2

Which further simplifies to:

x - 5 ≥ 3

Now, we add 5 to both sides to isolate x:

x ≥ 3 5

This simplifies to:

x ≥ 8

Therefore, the solution to the inequality 2x - 5 ≥ 11 is x ≥ 8.

Related Concepts and Tricks

Algebraic Solution Steps Recap

To summarize the steps used to solve the inequality:

Subtract the constant term on the left side from both sides to simplify the inequality. Divide both sides by the coefficient of x to isolate x. Add the constant to any term on the right side to isolate the variable x.

Mathematical Problem Solving

When solving inequalities, it's important to follow a systematic approach. The key is to isolate the variable on one side of the inequality. Remember to reverse the inequality sign when dividing or multiplying both sides by a negative number.

Additional Practice

To solidify your understanding, practice solving other linear inequalities. For instance, try solving 3x - 7 ≥ 16 or 4x 2 ≥ 18.

Conclusion

Solving the inequality 2x - 5 ≥ 11 involves a series of algebraic manipulations to isolate the variable x. The steps include subtracting a constant, dividing by the coefficient, and adding another constant to isolate the variable. By following these steps, you can solve similar inequalities efficiently.