Problem 2 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(11-c \leq 8\)

Step-by-Step Solution

Answer

The solution is \(c \geq 3\), with a graph of a closed dot at 3 and shading to the right.

Step 1: Isolate 'c' on One Side

To solve the inequality \(11-c \leq 8\), we need to isolate \(c\) on one side. Start by subtracting 11 from both sides to get the inequality in the form \(-c \leq 8 - 11\), which simplifies to: \(-c \leq -3\).

Step 2: Solve for 'c'

Now, we need to get \(c\) alone. Since we have \(-c \leq -3\), multiply both sides of the inequality by -1. Remember that when you multiply or divide by a negative number, you need to reverse the inequality sign. The result is: \(c \geq 3\).

Step 3: Number Line Representation

Draw a number line, and mark the point at \(c = 3\). Since the inequality is \(c \geq 3\), draw a closed dot on 3 and shade the line extending to the right to indicate all values greater than or equal to 3.

Key Concepts

Graphing Inequalities

Visualizing inequalities on a graph is one of the most effective ways to understand their solutions. Unlike equations, which show values that make both sides equal, inequalities represent a range of possible solutions. This is why graphing them provides a clearer picture.

When graphing inequalities like the one in this exercise, first determine the solution range. In this case, after solving, you found that \( c \geq 3 \). This means that the boundary of the solution set includes all numbers greater than or equal to 3.

Here's what you need to consider when graphing:

• Identify the boundary point, here it's 3.
• Decide whether the boundary is included or not; use a closed dot for \( \geq \) or \( \leq \), and an open dot for \( > \) or \( < \).
• Shade the portion of the graph that represents the solution set. For \( c \geq 3 \), this means shading to the right.

Graphing inequalities helps students visualize and verify the solution set effectively.

Solving Inequalities

Solving inequalities is similar to solving equations, with a few key differences. The main objective is to isolate the variable, but with inequalities, you must pay attention to certain rules, especially when multiplying or dividing by negative numbers.

For example, in the problem \( 11 - c \leq 8 \), you begin by isolating \( c \). The first step involves subtracting 11 from both sides to simplify to \(-c \leq -3\). When isolating a variable like \( c \), you must multiply both sides by -1, which flips the inequality sign, resulting in the final inequality: \( c \geq 3 \).

Essential points to remember:

• Always perform the same operation on both sides of the inequality to maintain its balance.
• Remember to reverse the inequality sign when multiplying or dividing by a negative number.

These steps ensure that you solve the inequality correctly and can confidently determine the range of solutions.

Number Line Representation

Representing solutions on a number line is a straightforward and effective visual method to display the solution of an inequality. It helps to clearly show which numbers satisfy the condition of the inequality.

In this exercise, after solving the inequality, your solution is \( c \geq 3 \). To represent this on a number line:

• First, draw a straight horizontal line.
• Place regular tick marks to represent numbers. Ensure these are evenly spaced.
• Identify the point corresponding to the solution boundary, 3 in this case.
• Since 3 is included in the solution set \( c \geq 3 \), use a closed dot on the number 3.
• Shade the line to the right of 3, indicating all numbers greater than or equal to 3.

Utilizing a number line not only shows the solutions at a glance but also reinforces understanding of how inequalities work. It visually demonstrates the continuum of possible solutions for the inequality.