Problem 9 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 2 x+6 \geq 10 $$

Step-by-Step Solution

Answer

In interval notation, the solution is \([2, \infty)\).

Step 1: Understand the Problem

We need to solve the inequality \( 2x + 6 \geq 10 \) for \( x \) and express the solution set using either set-builder notation or interval notation.

Step 2: Isolate the Variable Term

Begin by isolating the term containing the variable \( x \). This is achieved by subtracting 6 from both sides of the inequality: \[2x + 6 - 6 \geq 10 - 6\]which simplifies to:\[2x \geq 4\]

Step 3: Solve for the Variable

Now, solve for \( x \) by dividing both sides of the inequality by 2:\[\frac{2x}{2} \geq \frac{4}{2}\]which simplifies to:\[x \geq 2\]

Step 4: Express the Solution in Notation

Express the solution \( x \geq 2 \) in the desired notation.In interval notation, the solution is: \[[2, \infty)\]In set-builder notation, the solution set is: \[\{ x \mid x \geq 2 \} \]

Key Concepts

Set-Builder Notation

Set-builder notation is a great way to express the solution to an inequality in a compact form. It describes the qualities that elements must have to be included in the set. For example, if we've solved an inequality and found that \( x \) can be 2 or any number greater than 2, we express it as: \( \{ x \mid x \geq 2 \} \). The vertical bar \( "\mid" \) can be read as "such that." This reads as "the set of all \( x \) such that \( x \) is greater than or equal to 2." It provides a powerful, flexible way to define sets based on properties or conditions.

• Make sure the condition described follows the original inequality solution.
• It clearly indicates what values are included.
• In this form, the set-builder notation can communicate complex sets much more clearly.

Interval Notation

Interval notation is another method used to represent sets of numbers, particularly solutions to inequalities. It's a way to express the range of solutions in a concise form. For example, if we solve an inequality and find that the solution set of \( x \) is all numbers 2 and greater, this is written as: \([2, \infty)\). In interval notation:

• The bracket \([\) indicates that 2 is included in the solution set, meaning \( x \) can equal 2.
• The parenthesis \()\) next to the infinity symbol indicates that infinity isn't a number you reach, but that the solution extends indefinitely.

This form of notation provides clarity and shows open or closed intervals visually:

• Closed intervals include the endpoint values, using square brackets \([\text{ or } ]\).
• Open intervals do not include the endpoints, using parentheses \((\text{ or } )\).

Solving Inequalities

Solving inequalities is all about finding the set of values that satisfy a given inequality condition. The steps are similar to solving equations, with a few crucial differences. For example, let's solve: \( 2x + 6 \geq 10 \)

• Step 1: Isolate the variable term. Subtract 6 from both sides to get: \( 2x \geq 4 \).
• Step 2: Solve for the variable. Divide both sides by 2 to find: \( x \geq 2 \).

Always remember these key points when solving inequalities:

• If you multiply or divide the inequality by a negative number, flip the inequality sign.
• Check your solution by plugging numbers back into the original inequality to verify correctness.

Once solved, the solution can be expressed using either set-builder or interval notation, offering flexibility in representation.