Problem 11 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ x+5>1 $$

Step-by-Step Solution

Answer

The solution set in interval notation is \((-4, \infty)\).

Step 1: Isolate the Variable

Start with the inequality: \( x + 5 > 1 \).To isolate \( x \), subtract 5 from both sides:\[ x + 5 - 5 > 1 - 5 \]Simplifying:\[ x > -4 \]

Step 2: Graph the Inequality

To graph the inequality \( x > -4 \), draw a number line.Mark a point at \( -4 \). Use an open circle at \( -4 \) to indicate that \( -4 \) is not included in the solution set.Shade the line to the right of \( -4 \) to indicate that all numbers greater than \( -4 \) satisfy the inequality.

Step 3: Write the Solution in Interval Notation

The inequality \( x > -4 \) means that any value greater than \( -4 \) is part of the solution set.In interval notation, this is expressed as:\((-4, \infty)\). This indicates that the solution includes all real numbers greater than \( -4 \).

Key Concepts

Solving Inequalities

When solving inequalities, the goal is similar to solving equations: to isolate the variable. However, inequalities have a special rule when it comes to multiplying or dividing by a negative number. In our original problem, we are given the inequality \( x + 5 > 1 \). Our goal is to find the value of \( x \) that makes this inequality true.
To do this, we start by isolating \( x \). We subtract 5 from both sides of the inequality:

• \( x + 5 - 5 > 1 - 5 \)

Simplifying this gives:

• \( x > -4 \)

This tells us that \( x \) must be greater than \(-4\). Remember, since we're simply adding or subtracting here, no special rules concerning the inequality direction are applied. That's it for solving—simple arithmetic allows us to find which values of \( x \) satisfy the condition.

Graphical Representation

Graphing an inequality is a great way to visualize the solution set. For the inequality \( x > -4 \), our first task is to draw a number line. Here's how to visually represent it:

• First, mark the point \(-4\) on the number line.
• Next, since \(-4\) is not included in the solution set (because it's greater than \(-4\), not equal to), use an open circle at \(-4\) to denote this.
• Finally, shade the line to the right of \(-4\). This indicates all numbers greater than \(-4\) are valid solutions.

The open circle signifies that our solution starts just above \(-4\), and the shaded line continues to the right, showing all values greater than \(-4\). This visual representation makes it much easier to understand which numbers make the inequality true.

Interval Notation

Once you've solved an inequality and possibly graphed it, it's useful to express the solution in interval notation, which is a simple and compact form. For the inequality \( x > -4 \), we know that any value greater than \(-4\) is a solution.In interval notation, we express this as \((-4, \infty)\):

• The round bracket "(" at \(-4\) indicates that \(-4\) itself is not included in the solution, matching our use of an open circle in the graph.
• \(\infty\) signifies that the values continue infinitely in the positive direction, again using a round bracket to indicate that infinity is a concept rather than a reachable number.

Interval notation is very efficient for writing solutions, especially for inequalities that span a range of values. It succinctly represents all numbers in a continuous interval, using brackets to denote inclusion or exclusion.