Problem 320 - Elementary Algebra

Question

In the following exercises, solve. Nataly is considering two job offers. The first job would pay her \(\$ 83,000\) per year. The second would pay her \(\$ 66,500\) plus \(15 \%\) of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?

Step-by-Step Solution

Answer

Nataly's total sales need to be over \$110,000\.

Step 1: Express the total salary for the second job

Let the total sales be represented by the variable \( S \). The salary from the second job can be expressed as \(\text{Salary} = 66500 + 0.15 \cdot S \).

Step 2: Set up an inequality

To determine when the salary from the second job is higher than the salary from the first job, set up the inequality: \( 66500 + 0.15 \cdot S > 83000 \).

Step 3: Solve the inequality

Subtract 66500 from both sides of the inequality: \( 0.15 \cdot S > 83000 - 66500 \). This simplifies to \( 0.15 \cdot S > 16500 \). Next, divide both sides by 0.15 to solve for \( S \): \(\frac{16500}{0.15} \).

Step 4: Calculate the result

Perform the division to find the value of \( S \): \( S > \frac{16500}{0.15} = 110000 \).

Key Concepts

linear inequalities

A linear inequality is an expression that shows the relationship of one quantity being larger or smaller than another.
Unlike equations, which use the equal sign, inequalities use symbols such as <, >, ≤, or ≥.
In solving linear inequalities, you often perform similar steps as solving equations, such as addition, subtraction, multiplication, or division.
However, one key difference is when you multiply or divide by a negative number, you must reverse the inequality sign.
Understanding and solving linear inequalities is crucial in many real-world scenarios, like comparing job offers.

algebraic expressions

In the given exercise, we deal with algebraic expressions; an algebraic expression is a mathematical phrase that can include numbers, variables, and operators.
For example, in this case, the expression for the second job's salary is \( 66500 + 0.15S \).
Here, 66500 and 0.15 are coefficients, and S is the variable representing total sales.
When working with algebraic expressions, it's essential to follow the order of operations (PEMDAS/BODMAS) to simplify or solve them correctly.
Combining like terms and using properties of operations can help break down more complex expressions into simpler forms.

problem-solving steps

To solve the problem of determining when Nataly's second job offer becomes more lucrative, we can follow systematic problem-solving steps.
Here’s a deeper look into the steps:
1. **Express the total salary for the second job:** Let the total sales be represented by the variable \( S \). Thus, the salary from the second job can be written as \( 66500 + 0.15 \cdot S \).
2. **Set up an inequality:** We need to find when this salary is greater than the first job's salary, which is \( 83000 \). This leads to: \( 66500 + 0.15 \cdot S > 83000 \).
3. **Solve the inequality:** Subtract \( 66500 \) from both sides: \( 0.15 \cdot S > 16500 \). Next, divide by \( 0.15 \) to get: \( S > 110000 \).
By following these logical and structured steps, we get a clear solution that Nataly's sales need to exceed \( 110000 \) for her second job offer to be better.