Five more than three times a number is at least 20.
To solve this problem, we first assign a variable to the unknown number. Let the number be denoted by \(x\). The problem states "three times a number," which we translate as \(3x\). The phrase "five more than" this value means adding 5 to it, resulting in the expression \(3x + 5\). Finally, the phrase "is at least 20" tells us that this expression must be greater than or equal to 20, represented by the inequality \(\ge 20\). Putting it all together, we have the linear inequality \(3x + 5 \ge 20\).
Now that we have the inequality \(3x + 5 \ge 20\), our goal is to isolate the variable \(x\). We begin by subtracting 5 from both sides of the inequality to remove the constant term on the left:
\[3x + 5 - 5 \ge 20 - 5\]
\[3x \ge 15\]
Next, we solve for \(x\) by dividing both sides of the inequality \(3x \ge 15\) by 3. Since we are dividing by a positive number, the direction of the inequality sign remains unchanged:
\[\frac{3x}{3} \ge \frac{15}{3}\]
\[x \ge 5\]
Thus, the condition is satisfied by all real numbers greater than or equal to 5.
In real life, inequalities help us set thresholds or requirements. For example, if you are planning to rent a banquet hall that costs a base fee plus a per-person charge, you could use an inequality to determine the minimum number of guests you need to invite to justify the total expense or to stay within a specific budget. By solving \(3x + 5 \ge 20\), you are essentially finding the minimum threshold required to meet a specific goal.