Problem 54 - College Algebra

Question

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$3 \leq 4 x-3<19$$

Step-by-Step Solution

Answer

The solution to the inequality \(3 \leq 4x -3 <19\) is \(1.5 \leq x < 5.5\) or in interval notation: \([1.5, 5.5)\)

Step 1: Add 3 to All Parts

Add 3 to each part of the inequality to isolate the term with the variable: \(3 + 3 \leq 4x - 3 + 3 <19 + 3\). This simplifies to \(6 \leq 4x < 22\).

Step 2: Divide by 4

To further isolate the variable \(x\), divide all parts of the inequality by 4: \(\frac{6}{4} \leq \frac{4x}{4} < \frac{22}{4}\). This simplifies to \(1.5 \leq x < 5.5\). This is the solution in inequality form.

Step 3: Graph on Number Line

To graphically represent the solution set on a number line, draw a line and mark the points corresponding to the values \(1.5\) and \(5.5\). Since the inequality on the left is 'less than or equal to', the point at \(1.5\) is filled or closed indicating that \(1.5\) is included in the solution set. The inequality on the right is 'less than', so the point at \(5.5\) is open or unfilled, indicating that this endpoint is not included in the solution set.

Step 4: Express in Interval Notation

Interval notation is a simplified way of expressing the solution set. The solution set of \(1.5 \leq x < 5.5\) is written in interval notation as \([1.5, 5.5)\). The square bracket indicates that \(1.5\) is included in the solution and the round bracket indicates that \(5.5\) is not included.

Key Concepts

Interval Notation

Interval notation offers a compact way to capture the solution of an inequality. It expresses the range of values that fulfill the inequality, using brackets to denote whether endpoints are included.

Here’s how it works:

• Square brackets \( [ ] \) mean that the endpoint is included in the solution. This is typically used when the inequality symbol is \( \leq \) or \( \geq \).
• Round brackets \( ( ) \) indicate that the endpoint is not included, associated with \( < \) or \( > \).

In our example, the inequality \( 1.5 \leq x < 5.5 \) is captured as \( [1.5, 5.5) \). The square bracket at 1.5 shows inclusion, and the round bracket at 5.5 shows exclusion.

Number Line Graph

Graphing the solution set on a number line makes it easier to visualize which values satisfy the inequality.

Here’s how to do it:

• Draw a horizontal line and mark important points, such as 1.5 and 5.5 in our case.
• Use a solid dot or mark at 1.5 to show it is included in the solution.
• Use an open circle at 5.5 to show it is not part of the solution.
• Shade the region between the two points to indicate all numbers in this range satisfy the inequality.

This graphical representation makes the solution intuitive and straightforward to understand.

Solving Inequalities

Solving inequalities involves finding all the values that make the inequality true.

We follow a systematic approach:

• First, simplify the expression to isolate the term with the variable. In our example, add 3 to all sides of the inequality to get \( 6 \leq 4x < 22 \).
• Next, perform operations like division to isolate the variable completely, leading to \( 1.5 \leq x < 5.5 \).
• It's crucial to maintain the inequality's direction unless multiplying or dividing by a negative number.

By carefully applying these steps, you can solve most linear inequalities confidently.

Algebra

Algebra serves as the foundation for solving inequalities, allowing us to manipulate expressions and equations in a logical way.

Basic algebraic rules involved include:

• Adding or subtracting the same number from both sides to maintain equality, as seen when adding 3 to simplify \( 3 \leq 4x - 3 < 19 \).
• Dividing or multiplying all parts by the same number to isolate the variable, ensuring you keep the inequality consistent.
• Knowing how to interpret signs and solve accordingly without changing the solution's validity.

Mastering these concepts facilitates the solution of various mathematical problems and strengthens problem-solving skills.