Problem 28 - Prealgebra

Question

Solve each inequality. Then graph the solution on a number line. $$-13 \geq x-8$$

Step-by-Step Solution

Answer

The solution is \(x \leq -5\) and it's graphed with a solid circle at \(-5\) and shading to the left.

Step 1: Isolate the Variable

We start with the inequality \(-13 \geq x - 8\). To isolate \(x\), we need to get rid of the \(-8\) on the right side. We do this by adding 8 to both sides of the inequality. The modified inequality is then:\[-13 + 8 \geq x - 8 + 8\] Simplifying both sides:\[-5 \geq x\] or equivalently \(x \leq -5\).

Step 2: Interpret the Inequality

The inequality \(x \leq -5\) means we are looking for all \(x\) values that are less than or equal to \(-5\). This tells us that \(-5\) is the largest value \(x\) can be, and any number smaller than this is also a solution.

Step 3: Graph the Solution on a Number Line

To represent the solutions \(x \leq -5\) on a number line: 1. Draw a number line. 2. Locate the point \(-5\) and mark it clearly. 3. Since the inequality includes \(-5\) (denoted by the 'less than or equal to' sign), draw a filled or solid circle at \(-5\) to include this point. 4. Shade the portion of the number line to the left of \(-5\) to indicate all values smaller than \(-5\) are solutions.

Key Concepts

Isolating the Variable

When we solve an inequality, our goal is to get the variable by itself on one side of the inequality sign. This process is known as "isolating the variable." For the inequality \(-13 \geq x - 8\), the variable is \(x\).
First, we need to remove any numbers on the same side as \(x\) to clear the way for it to stand alone. In this case, we have to get rid of \(-8\) next to \(x\). We do this by performing the opposite operation. Since \(-8\) involves subtraction, we add 8 to both sides of the inequality to maintain balance.

• Start with: \(-13 \geq x - 8\).
• Add 8 to both sides: \(-13 + 8 \geq x - 8 + 8\).
• After simplifying both sides, this changes to \(-5 \geq x\).

This step reveals the inequality \(x \leq -5\), which shows all the possible values that the variable \(x\) can take.

Graphing Inequalities

Graphing an inequality on a number line is a great way to visualize the solution set. For the inequality \(x \leq -5\), here's how to represent it:
First, draw a horizontal line to serve as your number line.
On this line, find the point that corresponds to \(-5\).

• Mark \(-5\) with a filled-in circle. The filled circle indicates that \(-5\) is included in the solution set because of the "equal to" part in the \(\leq\) sign.
• Shade everything to the left of \(-5\) on the line. This shading shows all numbers that are less than \(-5\), which are part of the solution.

By highlighting the segment to the left, you communicate visually that any number in this region, including \(-5\) itself, satisfies the inequality.

Number Line

The number line is a basic but effective visual tool useful for understanding inequalities. It lays out numbers in order from left to right, representing all real numbers in one-dimensional space.
Each point on the line represents a specific number, allowing us to see the relationships between numbers quickly.

• A solid dot on a point signifies that the point itself is part of the solution.
• An open circle, on the other hand, would indicate that the point is not included in the set but the surrounding region is.
• The shaded region shows the range of solutions extending from the point.

Using the number line is beneficial because it offers an immediate, intuitive understanding of where solutions lie. It helps in grasping that for inequalities such as \(x \leq -5\), the solution set stretches infinitely to the left from \(-5\), encompassing smaller and more negative values.