Solve the given inequality for real\(x: \frac{1}{2}(\frac{3x}{5}+4) ≥ \frac{1}{3}(x-6)\).
Solve the given inequality for real\(x: \frac{1}{2}(\frac{3x}{5}+4) ≥ \frac{1}{3}(x-6)\).
Solution and Explanation
\(\frac{1}{2}(\frac{3x}{5}+4) ≥ \frac{1}{3}(x-6)\)
\(⇒ 3(\frac{3x}{5} + 4) ≥ 2(x - 6)\)
\(⇒ \frac{9x}{5} + 12 ≥ 2x - 12\)
\(⇒ 12 + 12 ≥ 2x - \frac{9x}{5}\)
\(⇒ 24 ≥ \frac{10x - 9x}{5}\)
\(⇒ 24 ≥ \frac{x}{5}\)
⇒ 120 ≥ x
Thus, all real numbers x, which are less than or equal to 120, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 120].
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Concepts Used:
Inequalities
In mathematics, inequality is a relationship that compares two numbers or other mathematical expressions in a non-equal fashion. It is most commonly used to compare the size of two numbers on a number line.
Specifically, a linear inequality is a mathematical inequality that integrates a linear function. One of the symbols of inequality is observed in a linear inequality: In graph form, it represents data that is not equal.
Some of the linear inequality symbols are given below:
- < less than
- > greater than
- ≤ less than or equal to
- ≥ greater than or equal to
- ≠ not equal to
- = equal to
Inequalities can be demonstrated as questions that are solved using alike procedures to equations, or as statements of fact in the form of theorems. It is used to contrast numbers and find the range or ranges of values that pleases a variable's criteria.