Verified Answer :

  • Subtract \(2x\) from both sides to group the variable \(x\): \(2x - 7 \geq 7\).
  • Add \(7\) to both sides to isolate the variable term: \(2x \geq 14\).
  • Divide both sides by \(2\) to find the solution: \(x \geq 7\).
  • The final solution is \(\boxed{x \geq 7}\).

Explanation

  1. Problem Analysis

The problem asks us to solve the linear inequality \(4x - 7 \geq 2x + 7\). Our goal is to isolate the variable \(x\) on one side of the inequality. This is a standard algebraic procedure where we use inverse operations to simplify the expression while maintaining the balance of the inequality. We are dealing with a linear inequality of the form \(ax + b \geq cx + d\), where \(a, b, c, d\) are constants. Our objective is to find the set of all real numbers \(x\) that satisfy this relationship.

  1. Collecting Variable Terms

To begin isolating \(x\), we first collect all terms containing \(x\) on the left side. We do this by subtracting \(2x\) from both sides of the inequality:
\[4x - 2x - 7 \geq 2x - 2x + 7\]
\[2x - 7 \geq 7\]

  1. Isolating the Variable Term

Next, we need to isolate the term containing \(x\) by removing the constant \(-7\) from the left side. We achieve this by adding \(7\) to both sides of the inequality:
\[2x - 7 + 7 \geq 7 + 7\]
\[2x \geq 14\]

  1. Solving for x

Finally, we solve for \(x\) by dividing both sides of the inequality by \(2\). Since \(2\) is a positive number, the direction of the inequality remains unchanged:
\[\frac{2x}{2} \geq \frac{14}{2}\]
\[x \geq 7\]
This result means that the inequality holds for all real numbers \(x\) that are greater than or equal to \(7\).

Examples

Imagine you have a budget to spend at a carnival. You want to play a game that costs \(\$2\) per round, but you also have to pay a one-time entry fee of \(\$7\). If you have a total of \(\$4\) per round to spend but must also account for a \(\$7\) penalty fee for using a special fast-pass, you can model your spending as \(4x - 7 \geq 2x + 7\), where \(x\) is the number of rounds. Solving this inequality helps you determine the minimum number of rounds you need to play so that your net gain or spending balance remains favorable. In this case, you need to play at least \(7\) rounds to satisfy the condition.

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