Problem 124 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The statement "the sum of \(x\) and \(6 \%\) of \(x\) is at least 80 " is modeled by \(x+0.06 x \geq 80\)

Step-by-Step Solution

Answer

The presented statement 'the sum of \(x\) and \(6\%\) of \(x\) is at least 80' is correctly modeled by the equation \(x + 0.06x \geq 80\). This statement is true.

Step 1: Understand the problem statement

The problem states that 'the sum of \(x\) and \(6\%\) of \(x\) is at least 80'. The word 'sum' in mathematics usually refers to the result of adding two or more numbers. The term '6\% of \(x\)' can be mathematically represented as \(0.06x\).

Step 2: Translate the statement into mathematical form

Translating 'the sum of \(x\) and \(6\%\) of \(x\) is at least 80' into mathematical form gives \(x + 0.06x \geq 80\). The word 'is' has been replaced by the symbol '\(\geq\)' to indicate 'at least', meaning greater than or equal to.

Step 3: Compare the translated form with the provided equation

The translated mathematical form \(x + 0.06x \geq 80\) matches with the provided equation. Thus, the given statement is correctly modeled by the equation.

Key Concepts

Understanding Inequalities

In mathematics, an inequality is like a comparison between two values or expressions, showing whether one is larger or smaller than the other. We use special symbols like '>', which means 'greater than,' or '<', which means 'less than.' For instance, the problem here uses the '≥' (greater than or equal to) symbol.
This symbol suggests that the expression on the left can either be larger than or exactly equal to the expression on the right. If we take a mathematical expression like \(x + 0.06x \geq 80\), it suggests the sum of \(x\) and \(6\%\) of \(x\) should be at least 80.

• If the expression on the left exactly equals 80, the overall statement still holds true.
• On the other hand, for any value greater than 80, the inequality remains valid.

This method proves useful when dealing with multiple solutions where one or more combinations can satisfy the given conditions.

Decoding Percentages

Percentages are a handy way to express parts of a whole as a number out of 100. When we say \(6\%\) of \(x\), it simply translates to \(0.06x\) mathematically.
This conversion from percentage to a decimal form makes calculations manageable and intuitive. The fraction \(\frac{6}{100}\) is directly equivalent to \(0.06\), an operation taught early in mathematics.

• Start by understanding that 'percent' means 'out of a hundred,' hence \(6\text{%}=\frac{6}{100}\).
• Every problem involving percentages can be transformed into a simple multiplication with decimals for easier understanding.
  So, \(6\%\) of \(x\) turns into \(0.06x\), making it easier to run arithmetic operations, such as addition or subtraction, as seen in our expression.

Frequently, percentages help present real-world problems, like financial calculations, in a much simpler format.

Applying Mathematical Modeling

Mathematical modeling takes real-world problems and translates them into mathematical expressions. This technique helps formulate problems so that they can be solved systematically.
In the exercise provided, we translated words into symbols and numbers, indicating the relationship using mathematics. This provided us with \(x + 0.06x \geq 80\), a clear model.

• First, dissect the problem statement—identifying quantities involved and their relationships.
• Next, assign these characteristics to a mathematical form, often aided by understanding symbols like '+', '-', and the inequality embeddings.

Through proper modeling, complex word problems become accessible. It teaches us to convert scenario-based terms, like 'at least 80' into mathematical language, ensuring a solvable structure. The precision needed in this process reinforces clarity and helps achieve desired results by using proper equations.