Problem 71 - Algebra for College Students

Question

Give, in interval notation, the unknown numbers in each description. One third of a number is added to 6 , giving a result of at least 3 .

Step-by-Step Solution

Answer

The unknown numbers are \([-9, \infty)\).

Step 1: Define the Variable

Let the unknown number be denoted as x.

Step 2: Set Up the Equation

One third of the number x is expressed as \(\frac{x}{3}\). Adding 6 to this gives us \(\frac{x}{3} + 6\).

Step 3: Inequality Formation

According to the problem, this sum is at least 3. So, we set up the inequality \(\frac{x}{3} + 6 \geq 3\).

Step 4: Isolate the Variable Term

To isolate the variable term, subtract 6 from both sides of the inequality: \(\frac{x}{3} + 6 - 6 \geq 3 - 6\), which simplifies to \(\frac{x}{3} \geq -3\).

Step 5: Solve for x

Multiply both sides of the inequality by 3 to solve for x: \(\frac{x}{3} \times 3 \geq -3 \times 3\), giving us \(x \geq -9\).

Step 6: Write the Solution in Interval Notation

The solution in interval notation is \([-9, \infty)\).

Key Concepts

interval notation

Interval notation is a way of writing subsets of the real number line. It is used mainly for expressing the solutions to inequalities. In interval notation, an interval is defined by its endpoints. The symbol \(( ... , ... )\) denotes an interval that does not include its endpoints, while \([ ... , ... ]\) denotes an interval that does include its endpoints. For instance, \([-9, \infty )\) represents all numbers from -9 up to positive infinity, including -9 but not \(\infty\). This is the interval format used to convey the solution to our inequality problem.
When writing intervals:

• Use parentheses \(( ... )\) when the endpoint is not included.
• Use brackets \([ ... ]\) when the endpoint is included.

Understanding how to read and write interval notation is essential for clearly articulating the solution sets of inequalities.

solving inequalities

Solving inequalities involves finding the set of all values of the variable that make the inequality true. Here are typical steps for solving an inequality:

• Define the variable: Assign a symbol (like x) to represent the unknown number.
• Set up the inequality: Translate the word problem into a mathematical inequality. For example, 'one third of a number added to 6 is at least 3' translates to \(\frac{x}{3} + 6 \geq 3\).
• Manipulate to isolate the variable: Use algebraic operations like addition, subtraction, multiplication, or division to get the variable alone on one side of the inequality. In our example, this would involve subtracting 6 and then multiplying by 3: \(\frac{x}{3} \geq -3\) -> \(x \geq -9\).
• Check for direction changes: Remember, multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Always check your steps to ensure accuracy.

By following these steps, the inequality \(\frac{x}{3} + 6 \geq 3\) simplifies to \(x \geq -9\).

algebraic expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (like +, -, *, /). In this particular exercise, our algebraic expression is \(\frac{x}{3} + 6\). To handle such expressions:

• Identify components: Break down the expression into parts—terms connected by addition or subtraction.
• Combine like terms: Simplify the expression by combining terms that have the same variable to the same power.
• Translate words into math: Understand word problems and translate them into algebra. For example, 'one third of a number' becomes \(\frac{x}{3}\).
• Solve step-by-step: Use a logical sequence of operations to simplify and solve. In our solution, we started by simplifying the inequality \(\frac{x}{3} + 6 \geq 3\) to isolate and solve for x.

Grasping algebraic expressions is vital as they form the basis for setting up equations and inequalities. By learning to manipulate these expressions, you can solve a wide range of algebra problems.