Problem 15 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ 2 c-1 < -5 \text { or } 3 c+2 \geq 5 $$

Step-by-Step Solution

Answer

The solution is \( c < -2 \) or \( c \geq 1 \).

Step 1: Solve the First Inequality

The first inequality is \( 2c - 1 < -5 \). To solve it, first add 1 to both sides: \( 2c < -4 \). Then, divide both sides by 2: \( c < -2 \). Thus, the solution to this inequality is \( c < -2 \).

Step 2: Solve the Second Inequality

The second inequality is \( 3c + 2 \geq 5 \). First, subtract 2 from both sides: \( 3c \geq 3 \). Then, divide both sides by 3: \( c \geq 1 \). Thus, the solution to this inequality is \( c \geq 1 \).

Step 3: Write the Compound Solution

The overall solution combines the solutions of both inequalities. The solution set is \( c < -2 \) or \( c \geq 1 \). This represents two separate intervals on the number line.

Step 4: Graph the Solution on a Number Line

To represent the solution on a number line, indicate all points less than -2 with an open circle at -2 and shading to the left. For the second inequality, indicate all points greater than or equal to 1 with a closed circle at 1 and shading to the right.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols like \(<, >, \leq, \text{or}\ \geq\). These symbols indicate a range of possible values that the variable can take. For example, in the inequality \(2c - 1 < -5\), the expression indicates that we are seeking values of \(c\) that make the expression less than \(-5\).

Steps to solve linear inequalities are much like those for solving equations. Start by isolating the variable on one side of the inequality:

• Perform addition or subtraction to remove constants.
• Use multiplication or division to solve for the variable.

However, remember that when you multiply or divide by a negative number, the inequality sign flips. This is crucial for maintaining the correct solution to the inequality. For our example, solving \(2c - 1 < -5\) led to \(c < -2\). Similarly, \(3c + 2 \geq 5\) yielded \(c \geq 1\).

Solution Set

After solving individual inequalities, like in our example, the solution is found by interpreting these results as a solution set. The solution set consists of all values that satisfy at least one of the inequalities.

In our case, the compound solution is \(c < -2\) or \(c \geq 1\). This indicates two separate intervals:

• \(c < -2\): Values to the left of \(-2\) on a number line.
• \(c \geq 1\): Values at \(1\) and to the right.

Each piece of the solution set represents a distinct inequality that could be true, emphasizing the word "or." This means any value in these intervals satisfies at least one inequality. Thus, either part of the solution set makes the original compound statement true.

Graphing

Graphing the solution set of inequalities helps visualize the range of potential solutions. By using a number line, we can clearly show which parts of the line are included in the solution.

For \(c < -2\), you would:

• Place an open circle at \(-2\), meaning this point is not included in the solution.
• Shade to the left, indicating all numbers less than \(-2\) are solutions.

For \(c \geq 1\), take these steps:

• Draw a closed circle at \(1\) since this number is included.
• Shade to the right, showing all numbers equal to or greater than \(1\) are part of the solution set.

This graphical representation allows for an immediate understanding of the intervals where the inequalities hold. It is a straightforward way to display solutions visually, ensuring that anyone can see all possible values that satisfy the inequalities.