Problem 80 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph the solution. \((\text {Lesson } 6.1)\) $$ -15>x-8 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -7\), and its graph is an open circle at -7, with a line extending to the left.

Step 1: Add 8 to both sides of the inequality

The first step is to isolate x. We can do this by adding 8 to both sides of the inequality to undo the subtraction of 8 from x. This gives us -15 + 8 > x - 8 + 8 which simplifies to -7 > x.

Step 2: Rewrite the inequality

The inequality -7 > x is equivalent to x < -7. So the solution to the inequality is all the values of x that are less than -7.

Step 3: Graph the solution

On the number line, label -7 and draw an open circle at -7 to indicate that -7 is not included in the solution. Draw a line to the left of -7, extending towards negative infinity to illustrate that the solution includes all numbers less than -7.

Key Concepts

Graphing Inequalities

When it comes to solving inequalities, graphing is a powerful visual tool that helps you better understand the range of solutions. In our exercise, we need to graph the inequality solution on a number line. First, let's consider what the inequality \( x < -7 \) means. It indicates that any value of \( x \) must be less than -7.
To graph the inequality, we need to:

• Label the point -7 on the number line.
• Place an open circle at -7. An open circle is crucial because it signifies that -7 is not included in the solution set (we use a closed circle if the number is included).
• Draw a line or arrow extending to the left from -7 since we are showing all numbers less than -7.

By using these steps, the graph will clearly represent the solution set for the inequality \( x < -7 \), helping you visualize all possible values that satisfy this condition.

Number Line

A number line is a simple yet effective way to represent solutions of inequalities visually. It consists of a horizontal line with markers at evenly spaced intervals, often including zero as a reference point.
The number line helps us quickly identify and compare different number values and relationships. In the context of inequalities:

• The position on the number line indicates the size of the number, with numbers increasing to the right and decreasing to the left.
• We use circles and lines/arrows to showcase which numbers are part of the solution set.
• An open circle means the number is not included in the solution. A closed circle represents inclusion.

For expressing \( x < -7 \), the open circle at -7 and the line extending to the left demonstrates all smaller values that satisfy the inequality. Thus, the number line serves as a valuable tool for understanding contexts and solutions involved in inequalities.

Algebraic Manipulation

Algebraic manipulation involves moving and rearranging terms in an equation or inequality to isolate the variable we are interested in solving for. It requires a step-by-step approach that maintains the equality or inequality's original condition. Let's break down how this is done:
To solve \(-15 > x - 8\), we wanted to find \(x\) by itself on one side of the inequality. The process includes:

• Adding or subtracting the same value from both sides to simplify the expression. For example, we added 8 to both sides to eliminate the \(-8\) attached to \(x\).
• Simplifying each side of the equation until \(x\) stands alone. This means doing the math, as in \(-15 + 8 = -7\).
• Rewriting the inequality to make it easier to interpret, swapping sides if needed. That's why we turned \(-7 > x\) into \(x < -7\), making it clearer for graphing purposes.

Understanding algebraic manipulation in inequalities is vital, as it gives you the means to work through any inequality problem to find the solution set effectively. It's all about maintaining balance and being methodical in your steps!