Verified Answer :

  • Identify the given inequality as \(4x \leq 20\).
  • Divide both sides of the inequality by 4 to isolate \(x\), noting that the inequality direction is preserved because 4 is a positive number.
  • Simplify the resulting expression to obtain the solution \(x \leq 5\).
  • State the final result: \(\boxed{x \leq 5}\).

Explanation

  1. Problem Analysis

The problem asks us to determine the possible values for \(x\) given the linear inequality \(4x \leq 20\). Here, \(x\) is a real variable that satisfies the condition that 4 times its value must be less than or equal to 20. To solve this, we need to isolate the variable \(x\) on one side of the inequality.

  1. Solving the Inequality

To isolate \(x\), we divide both sides of the inequality \(4x \leq 20\) by the coefficient of \(x\), which is 4. According to the properties of inequalities, when dividing both sides by a positive number, the direction of the inequality sign remains unchanged. Therefore, we perform the following calculation:
\[ \frac{4x}{4} \leq \frac{20}{4} \]
Simplifying both sides, we get:
\[ x \leq 5 \]
This result indicates that any real number less than or equal to 5 satisfies the original inequality.

  1. Conclusion

The inequality \(4x \leq 20\) simplifies to \(x \leq 5\). This means that any value of \(x\) in the interval \((-\infty, 5]\) is a valid solution. The final answer is \(\boxed{x \leq 5}\).

Examples

Consider a scenario where you have a budget of \$20 for printing flyers. If each flyer costs \$4 to print, you want to know how many flyers, represented by \(x\), you can print without exceeding your budget. The inequality \(4x \leq 20\) represents this constraint. Solving it gives \(x \leq 5\), meaning you can print at most 5 flyers. This helps in planning expenses and resource allocation effectively.

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