In the linear equation 9x − [5(2x + 1) / 2] = 9/2, find the value of x by simplifying and solving step by step.

Introduction / Context:

This question tests your ability to simplify and solve a basic linear equation that involves brackets and a fractional coefficient. Such equations are very common in aptitude tests and school level algebra, and the key skill is to remove fractions carefully and isolate the variable x without making sign or arithmetic mistakes.

Given Data / Assumptions:

• The equation is 9x − [5(2x + 1) / 2] = 9/2.
• x is a real number.
• Standard rules of arithmetic and algebra apply.
• The goal is to find the unique solution for x.

Concept / Approach:

We first remove the bracket and the fraction by expanding 5(2x + 1) and then clearing the denominator 2. After that, we collect like terms on one side of the equation and solve the resulting simple linear equation. Working with fractions is easier if we multiply the entire equation by the common denominator at an early stage.

Step-by-Step Solution:

Verification / Alternative check:

Substitute x = 7/4 into the original equation. Compute 2x + 1 = 2 * 7/4 + 1 = 7/2 + 1 = 9/2. Then 5(2x + 1) / 2 = 5 * (9/2) / 2 = 45/4. Also 9x = 9 * 7/4 = 63/4. So the left side is 63/4 − 45/4 = 18/4 = 9/2, which matches the right side. This confirms that x = 7/4 is correct.

Why Other Options Are Wrong:

The values −7/4, 4/7, −4/7, and 3/4 all fail when substituted back into the equation, giving left hand sides that are not equal to 9/2. These usually come from sign errors, incorrect clearing of fractions, or wrong division at the final step.

Common Pitfalls:

Learners often forget to multiply every term by 2 when clearing the denominator, or they expand 5(2x + 1) incorrectly. Another frequent mistake is dividing 14 by 8 but not simplifying the fraction properly. Taking each step slowly and checking arithmetic avoids these issues.

Final Answer:

The correct value of x that satisfies the equation is 7/4.