Problem 43 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$1-\frac{x}{2}>4$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -6\), expressed in interval notation as \((-\infty, -6)\). The graph on the number line displays an open circle at -6 with shading to its left indicating all numbers less than -6.

Step 1: Simplify the Inequality

First, simplify the inequality to isolate \(x\) on one side by subtracting 1 from both sides of the inequality. This gives: \(-\frac{x}{2} > 4 - 1\), which simplifies to: \(-\frac{x}{2} > 3.

Step 2: Solve for x

Next, multiply both sides by -2 to solve for \(x\). Remember that when you multiply or divide by a negative number when working with inequalities, the direction of the inequality sign must be reversed. This gives: \(x < -6\).

Step 3: Express Solution Set in Interval Notation

To present this solution set in interval notation, which includes all real numbers less than -6, use the format: \((-\infty, -6)\). This indicates that the solution includes all numbers up to but not including -6.

Step 4: Graph the Solution Set on the Number Line

For the graph on the number line, place an open circle at -6 (since -6 is not included in the solution) and shade everything to the left of -6.

Key Concepts

Interval Notation

Interval notation is a way to represent a range of values, like the solution to a linear inequality. For the inequality we've solved, where \(x < -6\), interval notation helps us express that all numbers less than -6 are part of the solution.

Here's how to read and write interval notation:

• Use parentheses, \((\), to indicate that an endpoint is not included. In our case, -6 is not included, so we write \((-\infty, -6)\).
• Infinities always have parentheses because they aren't actual numbers we can reach.

This format clearly tells us where our solutions start and end without including the boundaries.

Solution Set

A solution set is the collection of all possible values that satisfy an inequality. In this problem, solving the linear inequality \(1 - \frac{x}{2} > 4\) gives us a solution set where \(x\) is less than -6.

Understanding solution sets is crucial because:

• The set defines which values make the inequality true.
• It is often expressed in interval notation for clarity.

So, when you determine that \(x < -6\), the complete set of values fulfilling this condition is \((-\infty, -6)\). This set includes all numbers smaller than -6.

Number Line

A number line graph visually represents the solution set of an inequality. By using a number line, you can easily communicate which numbers are solutions without ambiguity.

For our inequality \(x < -6\):

• Place an open circle on the number line at -6. This circle indicates that -6 is not included in the solution set.
• Shade everything to the left of -6 to show all numbers less than -6 are included.

This visual helps in understanding which parts of the number line represent the solution, reinforcing the information given by interval notation.

Algebra

Algebra allows us to manipulate equations and inequalities to find unknown values. In this problem, algebra helps us solve the inequality by isolating \(x\).

Here's the process we used:

• First, simplify by moving terms to one side, which turned \(1 - \frac{x}{2} > 4\) into \(-\frac{x}{2} > 3\).
• Then, solve for \(x\) by multiplying both sides by -2, remembering to flip the inequality sign, giving \(x < -6\).

Understanding algebraic techniques is essential as they provide the tools to solve for variables and interpret results effectively. By mastering these skills, you can handle not only inequalities but a wide range of mathematical problems.