Question

create an inequality to represent each problem. be sure to include the following:

• define all variables.
• write each inequality.

1. a number is less than one - fourth the sum of eight times the number and four.
2. negative three times a number plus four is no more than the number minus eight.
3. a realtor is paid a base salary of $20,000 a year plus 10% commission on all property sales. what are the sales needed to ensure an annual income greater than $50,000?
4. a car salesperson is paid a base salary of $30,000 a year plus 9% commission on all sales. what are the sales needed to ensure an annual income greater than $75,000?

Explanation:

Problem 1

Step1: Define variable

Let the number be \( x \).

Step2: Translate the problem

"One - fourth the sum of eight times the number and four" is \( \frac{1}{4}(8x + 4) \). The problem states that a number (\( x \)) is less than this value. So the inequality is \( x<\frac{1}{4}(8x + 4) \).

Step1: Define variable

Let the number be \( x \).

Step2: Translate the problem

"Negative three times a number plus four" is \( - 3x+4 \), and "the number minus eight" is \( x - 8 \). The phrase "is no more than" means \( \leq \). So the inequality is \( -3x + 4\leq x-8 \).

Step1: Define variable

Let \( x \) be the total property sales (in dollars).

Step2: Calculate income components

The base salary is \( \$20,000 \), and the commission is \( 10\% \) of \( x \), which is \( 0.1x \). The total income is the sum of the base salary and the commission, and we want this total income to be greater than \( \$50,000 \). So the inequality is \( 20000+0.1x>50000 \).

Answer:

Let \( x \) be the number. The inequality is \( x<\frac{1}{4}(8x + 4) \)

Problem 2