Problem 8 - College Algebra

Question

In Exercises 1-12, graph the solutions of each inequality on a number line. $$x \leq 7$$

Step-by-Step Solution

Answer

The solution of the inequality \(x \leq 7\) is graphed by shading the portion of the number line to the left of and including 7.

Step 1: Understand the inequality

The inequality is \(x \leq 7\). This means that we are looking for all values of \(x\) that are less than or equal to 7.

Step 2: Mark the number 7 on the number line

Draw a number line and clearly mark the number 7. This will be the boundary point of the inequality shade.

Step 3: Indicate the solution set

The inequality \(x \leq 7\) indicates that any number less than or equal to 7 is a solution. Therefore, to graphically represent this, shade or color the portion of the number line to the left of (and including) 7.

Key Concepts

Number Line

A number line is a straight horizontal line that visually represents numbers in their order, allowing us to see their relationships and perform various mathematical operations. Think of it as a ruler with numbers placed at equal intervals, starting from a zero point and extending infinitely in both directions. On this line:

• Each point corresponds to a number.
• Positive numbers lie to the right of zero, while negative numbers are to the left.
• It helps in understanding the position and size of different values, including how they relate to one another.

When working with inequalities like the one given (\(x \leq 7\)), the number line becomes an essential tool for visualizing the range of possible values that 'x' can take.

Graphing Solutions

Graphing solutions involves illustrating the range of values that satisfy an inequality. In our case, we have the inequality \(x \leq 7\). This process uses the number line as the base for this graphical representation.

• The first step is to draw a number line and place the point 7 on it.
• Since the inequality includes numbers less than or equal to 7, you need to shade or highlight the section of the line extending to the left of 7, all the way to negative infinity.
• It's crucial to include 7 in this shaded area because the inequality is 'less than or equal to.'

The shaded section clearly shows all possible values of 'x' that satisfy the inequality, offering a quick and easy-to-understand solution at a glance.

Boundary Point

The boundary point is a critical concept when dealing with inequalities. For the inequality \(x \leq 7\), the number 7 acts as the boundary point.

• This point determines where the solution set starts or ends on the number line.
• When the inequality includes equality (as in \( \leq \) or \( \geq \)), the boundary point is usually depicted with a closed circle on the graph.
• If the inequality does not include equality (like \( < \) or \( > \)), an open circle is used.

In our example, because \(x \leq 7\) includes the equal sign, a closed circle is drawn around the number 7 on the number line to signify that 7 is part of the solution set.

Solution Set

The concept of a solution set is at the heart of understanding inequalities like \(x \leq 7\). A solution set includes all values that satisfy a given inequality, forming a complete set of potential answers.

• For our inequality, the solution set comprises all real numbers that are less than or equal to 7.
• In practice, this means starting at 7 and going infinitely to the left on the number line.
• The boundary point (7) indicates where this set begins, including this value due to the 'equal to' part of the inequality.

Graphically, the solution set is easy to see through the shaded area on the number line, which visually encapsulates all the numbers that satisfy the condition given in the inequality.