Verified Answer :

  • Begin with the inequality \(13 < 2x - 1\).
  • Add 1 to both sides to get \(14 < 2x\).
  • Divide by 2 to isolate \(x\), resulting in \(7 < x\).
  • The solution is the set of all \(x > 7\), which is \(\boxed{x > 7}\).

Explanation

  1. Analysis of the Inequality

The problem asks to find the range of all real numbers \(x\) that satisfy the linear inequality \(13 < 2x - 1\). This inequality involves a single variable \(x\) and constants. Our objective is to isolate \(x\) on one side of the inequality to determine the set of values that fulfill the given condition. We will perform algebraic manipulations to achieve this isolation.

  1. Isolating the Term with \(x\)

To solve for \(x\), we first isolate the term containing \(x\). We begin by adding \(1\) to both sides of the inequality:
\[13 + 1 < 2x - 1 + 1\]
\[14 < 2x\]
This step simplifies the expression by removing the constant term on the right-hand side.

  1. Solving for \(x\) and Final Conclusion

Next, we divide both sides of the inequality by \(2\) to solve for \(x\). Since \(2\) is a positive number, the direction of the inequality remains unchanged:
\[\frac{14}{2} < \frac{2x}{2}\]
\[7 < x\]
This result shows that any real number \(x\) greater than \(7\) will satisfy the original inequality. In interval notation, this is represented as \((7, \infty)\).

Examples

Consider a scenario where you are setting a minimum requirement for a test score or a weight threshold. If a system requires that twice your score minus 1 must be greater than 13, you need to score higher than 7. This math helps ensure that variables meet safety or performance standards, similar to how engineers calculate minimum load-bearing capacities for structures using linear constraints.

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