Problem 30 - College Algebra

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(8 x-2 \geq 14\)

Step-by-Step Solution

Answer

The solution to the inequality \(8 x-2 \geq 14\) is \(x \geq 2\), in interval notation this is \([2, \infty)\). The solution set is graphed on a number line with a filled dot on 2 and an arrow pointing to the right.

Step 1: Isolate the variable x in the inequality

Look at the inequality \(8 x-2 \geq 14\). The first step is to isolate the x variable. This can be done by firstly adding 2 to both sides of the inequality in order to remove the -2 from the left side of the inequality. This results in \(8 x \geq 16\). Then, divide both sides of the inequality by 8 to solve for x. This gives the inequality \(x \geq 2\).

Step 2: Express the solution in interval notation

The solution to the inequality \(x \geq 2\) can be expressed in interval notation. In interval notation, the solution is expressed as a range of values which x can take on the number line. In this case, x takes all values greater than or equal to 2. This is expressed in interval notation as \([2, \infty)\). The square bracket indicates that the number 2 is included in the solution set. The infinity symbol (\(\infty\)) is used to indicate that there is no upper limit to the values of x. The parenthesis is used because \(\infty\) is not a real number and x can't actually achieve this value, so it is not included in the solution set.

Step 3: Graph the solution on a number line

The solution set can be represented on a number line by marking the point at x=2 with a filled dot (to indicate that it is included in the solution set) and drawing an arrow to the right (to indicate that all values greater than 2 are also part of the solution set).

Key Concepts

Interval Notation

Understanding interval notation is crucial for expressing solution sets of inequalities. It provides a compact way to illustrate ranges of values. In the inequality solution process, we convert our final inequality into this format to clearly communicate which values of the variable satisfy the inequality.
In the example from the original exercise, the solution set for the inequality \(x \geq 2\) is expressed using interval notation as \([2, \infty)\). This format has a few distinct characteristics:

• The square bracket \([\)] means that 2 is included in our solution. It covers all values starting from 2 and moving to the right on the number line.
• The infinity symbol \((\infty)\) is used to denote that there is no upper limit. Since \(\infty\) isn't a real number, we use a parenthesis to show it isn't included in the set.

Interval notation is a concise and universal way to convey solutions, making it easier to interpret and understand.

Number Line Graph

A number line graph is a simple yet effective visual representation of inequality solutions. It helps in understanding which parts of the number line are included in the solution.
To graph the solution of the inequality \(x \geq 2\):

• Start by marking the point at \(x = 2\) with a filled dot. This indicates that the number 2 is included in the set of solutions.
• Draw a straight line extending to the right from the dot. This line shows that all numbers greater than 2 are also part of the solution set.
• An arrow at the end of the line indicates that the line continues indefinitely in that direction, aligning with the concept of extending to infinity.

Number line graphs are particularly useful for quickly checking the validity of solutions and understanding the range of possible values for a variable.

Inequality Solutions

Solving linear inequalities involves several key steps that help isolate the variable and identify its potential values.
The original exercise shows how to work through the inequality \(8x - 2 \geq 14\). By following these steps:

• **Isolate the variable**: Begin by reversing the operations around \(x\). Add 2 to both sides to get \(8x \geq 16\). Then divide each side by 8, yielding \(x \geq 2\).
• **Check your operations**: Ensure that each operation is performed correctly, especially when dividing or multiplying by negatives, as these can reverse the inequality sign.
• **Express solutions clearly**: After isolating the variable, express the solution in a clear form, such as interval notation. This ensures the solution is well-represented.

By understanding and applying these steps, you can confidently solve linear inequalities and express their solutions in an accurate format.