Verified Answer :

  • Analyze the inequality: \(3x - 7 > 5\).
  • Add 7 to both sides: \(3x > 12\).
  • Divide both sides by 3 to isolate \(x\): \(x > 4\).
  • The solution set is \(\boxed{x > 4}\).

Explanation

  1. Problem Analysis

The goal is to solve the linear inequality \(3x - 7 > 5\). We are dealing with a variable \(x\) and constants that we need to manipulate to isolate the variable. The inequality represents a relationship where one expression must be strictly greater than another. By applying standard algebraic rules for inequalities, we can determine the range of values for \(x\) that satisfy this condition.

  1. Isolating the Variable Term

To isolate the variable term \(3x\), we need to eliminate the constant \(-7\) from the left side of the inequality. We do this by adding \(7\) to both sides of the inequality:
\[3x - 7 + 7 > 5 + 7\]
\[3x > 12\]
Adding the same value to both sides of an inequality is an equivalent operation that preserves the direction of the inequality.

  1. Solving for \(x\)

Now that we have the inequality in the form \(3x > 12\), we can solve for \(x\) by dividing both sides by the coefficient of \(x\), which is \(3\). Since \(3\) is a positive number, the direction of the inequality remains unchanged:
\[\frac{3x}{3} > \frac{12}{3}\]
\[x > 4\]
Thus, the original inequality \(3x - 7 > 5\) is satisfied for all \(x\) greater than \(4\).

Examples

Linear inequalities are used in various real-world scenarios to represent constraints. For example, consider a budget constraint for a project. If you have already spent \(\$700\) on initial materials and each additional unit of work costs \(\$300\), and you want to ensure your total cost exceeds \(\$500\), you can model this as \(300x - 700 > 500\). Solving this helps you determine how many units of work must be performed to meet your target expenditure threshold. Here, \(300x > 1200\), so \(x > 4\) units of work are required.

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