Problem 5 - Algebra A Combined Function

Question

Graph the solution set of each inequality on a number line and then write it in interval notation. $$ \\{x \mid-7 \leq x\\} $$

Step-by-Step Solution

Answer

The solution set is \([-7, \infty)\).

Step 1: Understand the Inequality

The inequality is given in set-builder notation: \( \{x \mid-7 \leq x\} \). This reads as the set of all \( x \) such that \( -7 \leq x \). This means \( x \) can be any number greater than or equal to \(-7\).

Step 2: Represent the Inequality on a Number Line

To graph the solution set on a number line, we place a closed circle at \(-7\) because \( x \) can be equal to \(-7\). Then, since \( x \) can take any value greater than \(-7\), we shade the number line to the right of \(-7\). This shaded region represents all numbers greater than or equal to \(-7\).

Step 3: Express the Solution in Interval Notation

In interval notation, we express the solution set as \([-7, \, \infty)\). The square bracket \([-7\) indicates that \(-7\) is included in the solution set (due to the \( \leq \) inequality), and the parenthesis \(, \infty)\) indicates that there is no upper bound on \( x \).

Key Concepts

Set-Builder Notation

Set-builder notation is a way to describe a set of numbers or values that satisfy a particular condition. In our example, the inequality is given as \( \{x \mid -7 \leq x\} \). This notation tells us that we are looking at all possible values of \( x \) that are greater than or equal to -7.
This is read aloud as "the set of all \( x \) such that \( -7 \leq x \)."
Set-builder notation is particularly useful because:

• It succinctly describes complex sets.
• It clearly states the rule or condition that elements of the set should satisfy.
• It is a common mathematical language for representing conditions, making it widely understood.

Using set-builder notation helps visually and formally express constraints often encountered in inequalities.

Graphing on a Number Line

Graphing an inequality on a number line is an excellent way to visualize its solution set. To graph \( \{x \mid -7 \leq x\} \) on a number line, you need to identify all numbers that meet the inequality's conditions.
Here is a step-by-step guide to graph this inequality:

• Start by drawing a horizontal line that represents the number line.
• Locate the point \(-7\) on this number line.
• Place a closed circle on \(-7\) because the inequality includes \(-7\), as denoted by the "less than or equal to" symbol \(\leq\).
• Since \(x\) can be any number greater than \(-7\), shade the line extending to the right from \(-7\). The shading indicates that all those numbers are part of the solution set.

Visualizing an inequality on a number line can clarify which numbers satisfy it and highlight the endpoints' inclusion or exclusion.

Interval Notation

Interval notation is a concise way of expressing the set of numbers that represent the solution to an inequality. In the exercise, the interval notation for \( \{x \mid -7 \leq x\} \) is \([-7, \infty)\). Here's how interval notation works:

• The square bracket \([-7\) means that \(-7\) is included in the set because the inequality \(-7 \leq x\) allows \(x\) to equal \(-7\).
• The parenthesis \(\infty)\) indicates that there is no upper bound on \(x\), and \(\infty\) is never included because it is not a definite number.

Remember:

• Square brackets mean that the endpoint is included in the interval.
• Parentheses mean that the endpoint is not included.

Interval notation provides a streamlined way to describe a range of numbers without using words or lengthy explanations.