Bob is looking for internet service for his new apartment. If he continues with his old internet provider, he will pay \( \$ 65 \) per month and receive a \( \$ 50 \) customer loyalty credit. The new internet provider he is considering will charge a one-time installation fee of \( \$ 150 \), plus \( \$ 40 \) per month. After how many months would the two options cost the same?

To find out after how many months the two options would cost the same, we need to set up an equation representing the total cost of each option over time. Let's denote: - \( x \) as the number of months after which the costs will be the same. For the old internet provider: - Monthly cost = \( \$65 \) - Customer loyalty credit = \( \$50 \) - Total cost after \( x \) months = \( 65x - 50 \) For the new internet provider: - One-time installation fee = \( \$150 \) - Monthly cost = \( \$40 \) - Total cost after \( x \) months = \( 150 + 40x \) Now, we can set up the equation: \[ 65x - 50 = 150 + 40x \] Let's solve this equation to find the value of \( x \), which represents the number of months after which the costs will be the same. Solve the equation by following steps: - step0: Solve for \(x\): \(65x-50=150+40x\) - step1: Move the variable to the left side: \(65x-50-40x=150\) - step2: Subtract the terms: \(25x-50=150\) - step3: Move the constant to the right side: \(25x=150+50\) - step4: Add the numbers: \(25x=200\) - step5: Divide both sides: \(\frac{25x}{25}=\frac{200}{25}\) - step6: Divide the numbers: \(x=8\) After 8 months, the costs of the old internet provider and the new internet provider will be the same.