Problem 15 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(-8 p \geq 24\)

Step-by-Step Solution

Answer

The solution is \( p \leq -3 \) and is graphed with a solid circle on \(-3\) and shading to the left.

Step 1: Understand the Inequality

We start with the inequality \(-8p \geq 24\). Our goal is to isolate the variable \(p\). This inequality tells us that negative eight times some number \(p\) is greater than or equal to 24.

Step 2: Isolate the Variable

To isolate \(p\), divide both sides of the inequality by \(-8\). Remember, when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. This gives us \( p \leq -3\).

Step 3: Interpret the Solution

The solution \(p \leq -3\) means any number that is less than or equal to \(-3\) will satisfy the inequality.

Step 4: Graph the Solution on a Number Line

On a number line, represent \(p \leq -3\) by drawing a solid circle at \(-3\) to include \(-3\) itself, and shade the line to the left of \(-3\), extending to negative infinity. This indicates that values of \(p\) are less than or equal to \(-3\).

Key Concepts

Graphing Inequalities

Graphing inequalities is a visual method to show solution sets of inequalities. Inequalities allow us to compare two values, indicating if one is greater, less, or sometimes equal to the other. When graphing inequalities, the main aim is to display all potential values that satisfy the inequality condition.

• When the inequality includes "greater than or equal to" (≥) or "less than or equal to" (≤), the value itself is included in the solution set. This is represented by a closed or solid circle on a graph.
• For "greater than" (>) or "less than" (<), the solutions do not include the value precisely, so we use an open circle.

This kind of visual representation on the number line makes it much easier to see how many solutions exist and which segments of numbers are included. For instance, in the inequality \(-8p \geq 24\), the graph will emphasize all numbers 'p' that are less than or equal to \(-3\).

Number Line Representation

Using a number line to represent inequalities helps clarify which numbers satisfy the condition. It offers a clear picture of where these numbers lie relative to the chosen boundaries. Here's how you represent an inequality on a number line:

• Identify the critical point, or boundary, based on the inequality. For the inequality \(p \leq -3\), the critical point is \(-3\).
• If the inequality sign is \(\leq\) or \(\geq\), indicate the boundary on the number line by using a solid circle, since the point \(-3\) itself satisfies the condition, "less than or equal to \(-3\)."
• Shade the line starting from the solid circle and continuing towards the direction of the numbers that fulfill the inequality. Here, this would be shading to the left of \(-3\).

This representation provides an intuitive understanding of the solution set. It helps you see how the variable can take any value leftward from the point, including the point itself.

Inequality Transformation

Inequality transformation involves the manipulation of inequalities to isolate variables and find solutions. When dealing with inequalities, there are important rules to remember:

• Operations like addition, subtraction, multiplication, and division can be performed on both sides of an inequality, just like equations.
• A crucial point is that multiplying or dividing both sides by a negative number requires reversing the inequality sign. This is what occurred in the problem \(-8p \geq 24\), resulting in dividing by \(-8\) and reversing to \(p \leq -3\).

Being meticulous with these operations ensures that you maintain the correct direction of the inequality sign, leading to accurate solutions. This transformation helps effortlessly transition mathematical expressions and derive meaningful solutions to inequalities.