Problem 14 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(\frac{g}{-3} \geq-9\)

Step-by-Step Solution

Answer

The solution is \(g \leq 27\). Graph: Closed circle at 27 with arrow to the left.

Step 1: Isolate the Variable

Start by isolating the variable \(g\) in the inequality. The inequality given is \(\frac{g}{-3} \geq -9\). To isolate \(g\), multiply both sides of the inequality by \(-3\). This step requires flipping the inequality sign because multiplying or dividing by a negative number reverses the direction of the inequality. This results in \(g \leq 27\).

Step 2: Graph the Solution on a Number Line

To graph the solution \(g \leq 27\) on a number line, draw a number line with numbers marked at regular intervals. Mark a point and an arrow extending to the left from 27 to represent all numbers less than or equal to 27. Use a closed circle at 27 to indicate that 27 is included in the solution set.

Key Concepts

Solving Inequalities

When faced with understanding and solving inequalities, the key is to identify the steps needed to isolate the variable. Unlike regular equations, inequalities use symbols such as \(<, >, \leq, \geq\) to indicate a range of possible solutions. The process of solving these problems often involves:

• Performing operations that simplify or isolate the variable, much like solving equations.
• Being mindful of how operations affect the inequality sign. This is crucial if we multiply or divide by a negative number, as it reverses the inequality's direction.

For example, in solving the inequality \(\frac{g}{-3} \geq -9\), multiplying both sides by \(-3\) isolates \(g\) but also flips the sign, leading to \(g \leq 27\). This ensures the variable remains part of the solution set defined by the inequality.

Inequality Graphing

Graphing inequalities is a visual method of presenting solutions on a coordinate system. It's helpful since it provides a clear representation of where solutions lie on the number line. Here is a simplified breakdown of how to graph an inequality:

• First, convert your inequality into a simple form where the variable is isolated, as shown in solutions like \(g \leq 27\).
• Determine whether the inequality is strict (using \(<\) or \(>\)) or inclusive (using \(\leq\) or \(\geq\)). An inclusive inequality means the endpoint itself is included in the solution.

In our example, \(g \leq 27\) suggests all values of \(g\) up to and including 27 are solutions. This is portrayed on the graph by using a filled circle at 27, reinforcing that 27 is part of the solution.

Number Line Representation

Representing inequalities on a number line provides a straightforward, visual sense of the solution set. To represent an inequality like \(g \leq 27\) on a number line:

• Draw a horizontal line with evenly spaced marks representing several numbers.
• Find the critical value, 27, mark it with a point, and use a filled circle if 27 is a solution (or an open circle if it isn't). A closed circle indicates inclusion.
• Extend an arrow from 27 to the left to demonstrate that all numbers less than or equal to 27 are included in the solution set.

This clear representation helps in understanding how the solutions expand infinitely in one direction, making the solution easy to visualize and comprehend.