Problem 47 - Elementary Algebra

Question

Set up an algebraic inequality and then solve it. If five is added to three times a number, then the result is less than twenty.

Step-by-Step Solution

Answer

The number must be less than 5.

Step 1: Define the Variable

Let the unknown number be represented by the variable \( x \). This will be the number we need to solve for in the inequality.

Step 2: Set Up the Inequality

According to the problem, "five is added to three times a number," which translates to the expression \( 3x + 5 \). We are also told this result is less than twenty. So, we set up the inequality: \( 3x + 5 < 20 \).

Step 3: Subtract 5 from Both Sides

To isolate the term with the variable (\( 3x \)), subtract 5 from both sides of the inequality: \( 3x + 5 - 5 < 20 - 5 \). Simplifying gives \( 3x < 15 \).

Step 4: Divide by 3

To solve for \( x \), divide both sides by 3: \( \frac{3x}{3} < \frac{15}{3} \). This simplifies to \( x < 5 \).

Step 5: Solution Interpretation

The solution \( x < 5 \) means that the number in question must be less than 5 to satisfy the original inequality condition that adding five to three times the number yields a result less than twenty.

Key Concepts

Variable Definition

In algebra, the first step to solving a problem often involves defining a variable. A variable is a symbol, typically a letter like \( x \), that represents an unknown value in an equation or inequality. In our exercise, we're tasked with finding a number that, when manipulated according to the problem's conditions, yields a specific result. Thus, we define this unknown number as \( x \).

• Variables act as placeholders for numbers.
• They allow us to set up equations and inequalities to solve problems.

Choosing the right variable is crucial for setting the stage for the rest of the solution. Once we've defined our variable, we can move on to translating the problem statement into a mathematical form.

Inequality Setup

Once we've defined our variable, the next step is setting up the inequality. An inequality tells us about the relative size of values rather than giving an exact answer. In our example, we need to translate the sentence "five is added to three times a number, then the result is less than twenty" into a mathematical inequality.
To break it down:

• "Three times a number" becomes \( 3x \).
• "Added to five" translates to \( 3x + 5 \).
• "Less than twenty" sets up the inequality \( 3x + 5 < 20 \).

This inequality represents the relationship described in the problem between the variable \( x \) and the constants involved. It is key to understanding how adding and multiplying affect the outcome, all while comparing it to another number, in this case, twenty.

Solution Interpretation

Interpreting the solution of an inequality means understanding what the result signifies in terms of the original problem. Solving the inequality \( 3x + 5 < 20 \) gave us the result \( x < 5 \). Here's what this means:

• \( x < 5 \) tells us the set of values for \( x \) that satisfy the inequality.
• Any number less than 5, when substituted back into the original expression \( 3x + 5 \), will ensure that the result is less than 20.
• This means the solution set includes all real numbers less than 5, making \( x \) continuous up to but not including 5.

Interpreting solutions in context is crucial for understanding what an inequality tells us about the possible values of a variable. It highlights relationships and constraints, guiding us to meaningful answers in real-world scenarios.