Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(3 \leq 2 x-1<7\)

Step-by-Step Solution

Answer

The solution to the inequality is \(2 \leq x < 4\).

Step 1: Define the inequations

It's beneficial to split the compound inequality into two separate inequalities. They are \(3 \leq 2x - 1\) and \(2x - 1 < 7\).

Step 2: Solve the first inequality

Solve \(3 \leq 2x - 1\). Start by adding 1 to both sides of the inequality: \(4 \leq 2x\). Then, divide both sides by 2: \(2 \leq x\), or \(x \geq 2\).

Step 3: Solve the second inequality

Solve \(2x - 1 < 7\). Start by adding 1 to both sides of the inequality, it becomes \(2x < 8\), then divide both sides by 2, get \(x < 4\).

Step 4: Visualize the solution on the number line

Plot the solution \(2 \leq x < 4\) on the real number line. The number line starts at 2 and ends at 4. Because the inequality includes 2, there is a darkened or filled circle at 2. Since the inequality does not include 4, a circle with no fill is drawn at 4. The section of number line between these points will be shaded to indicate all the possible values of x.

Step 5: Formulate the final solution

The solution includes values greater than or equal to 2 but less than 4. Then the final answer is \(2 \leq x < 4\).

Key Concepts

Real Number Line

The real number line is a visual representation of all real numbers arranged in a straight line. Each point on this line corresponds to a real number, which includes all the numbers we use in everyday life: positive numbers, negative numbers, and zero. This line extends infinitely in both directions representing positive and negative infinity.

When graphing solutions to inequalities, the real number line is particularly useful. We use it to depict the range of values that satisfy our inequality conditions. By marking and shading parts of the line, we can communicate the interval of real numbers that are solutions to an inequality.

For instance, if the solution to an inequality is that a number must be strictly less than 4, we'd draw an open circle at 4 to show that 4 is not included in the solution set. Conversely, if the inequality allows for a number to be equal to, say, 2, we'd place a filled circle at 2 to indicate its inclusion.

Compound Inequalities

Compound inequalities combine two individual inequalities connected by the words "and" or "or." They help us understand the scope of acceptable solutions in a broader context than a single inequality. The solution to a compound inequality is a range of values that simultaneously satisfy both inequalities.

Consider the example of the inequality given: \(3 \leq 2x - 1 < 7\). This is a compound inequality using "and," meaning we're looking for values of \(x\) that satisfy both \(3 \leq 2x - 1\) and \(2x - 1 < 7\) at the same time.

To solve it, break down the compound statement into two separate inequalities. This gives: \(3 \leq 2x - 1\) and \(2x - 1 < 7\). Solve each inequality independently to determine the range of values for \(x\) that satisfy both conditions. This method streamlines the process while maintaining accuracy.

Graphing Inequalities

Graphing inequalities involves representing the solution set on a number line. This visual aid helps make sense of which values satisfy the inequality. Let's explore how to graph the compound inequality \(2 \leq x < 4\) using the real number line.

Begin by drawing a number line and identify the specific boundary points of the solution, which are 2 and 4 in this case. Since the inequality includes the number 2, you create a filled circle on the number line at 2. The filled circle indicates that this point is part of the solution set.

At 4, the inequality does not include this number, implying an open circle. Therefore, you draw an open circle at 4 to show that it is not included in the solution set. Then, shade the line between 2 and 4 to show all the possible values of \(x\) that lie within this interval.

• A filled circle corresponds to "greater than or equal to" (\(\geq\) or \(\leq\)).
• An open circle indicates "less than" or "greater than" (\(<\) or \(>\)).

This graphical representation clarifies the solution, visually explaining how the inequality constraints are applied. It's a powerful tool for verifying solutions and understanding the extent of possible values.