Question

You are running for class president. At 2: 30 on election day you have 95 votes and your opponent has 120 votes. Forty-five more students will be voting. Let \(x\) represent the number of students (of the 45) who vote for you. a. Write an inequality that shows the values of \(x\) that will allow you to win the election. b. What is the smallest value of \(x\) that is a solution of the inequality?

Short answer

To win the election, you need more than 35 of the remaining votes, so the smallest value of \(x\) that solves the inequality is 36.

Step-by-step solution

Define the Variables

Let \(x\) represent the number of students who vote for you out of the remaining 45 students. Your current vote total is 95 and your opponent's total is 120.

Write the Inequality

To win, you must have more votes than your opponent. You currently have 95 votes and you will add \(x\) votes from the remaining 45 students. Your opponent currently has 120 votes and will receive the remainder of the 45, which is 45 - \(x\). Therefore, the inequality that represents this situation is: 95 + \(x\) > 120 + (45 - \(x\)).

Solve the Inequality

Solve the inequality, 95 + \(x\) > 120 + 45 - \(x\), for \(x\). Add \(x\) to both sides to obtain: 95 + 2\(x\) > 165, then subtract 95 from both sides to get: 2\(x\) > 70. Finally, divide both sides by 2, to find that \(x\) > 35.

Identify Minimum Value

According to the inequality, \(x\) > 35, meaning you need more than 35 votes out of the remaining 45 to win. So, the smallest possible integer value for \(x\) that will satisfy the inequality is \(x\) = 36.