Problem 61 - College Algebra with Modeling and Visualization

Question

Solve the compound linear inequality graphically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth whenever appropriate. $$ 3 \leq 5 x-17<15 $$

Step-by-Step Solution

Answer

The solution in interval notation is \([4, 6.4)\).

Step 1: Break Down the Compound Inequality

The compound inequality is given as two separate inequalities combined together: \(3 \leq 5x - 17\) and \(5x - 17 < 15\). We will solve each part step by step.

Step 2: Solve the First Inequality

The first inequality is \(3 \leq 5x - 17\). Add 17 to both sides to get: \[3 + 17 \leq 5x\] which simplifies to \[20 \leq 5x\]. Next, divide both sides by 5: \[\frac{20}{5} \leq x\] so \(x \geq 4\).

Step 3: Solve the Second Inequality

The second inequality is \(5x - 17 < 15\). Add 17 to both sides to get: \[5x < 15 + 17\] which simplifies to \[5x < 32\]. Divide both sides by 5: \[x < \frac{32}{5}\] so \(x < 6.4\).

Step 4: Combine the Solution

Combining the results from Step 2 and Step 3, we have \(4 \leq x < 6.4\). This is the solution to the compound inequality, which states that \(x\) is greater than or equal to 4 and less than 6.4.

Step 5: Graph the Solution

To graph \(4 \leq x < 6.4\), plot a number line. Draw a solid circle at 4 to indicate that 4 is included in the solution set. Draw an open circle at 6.4 to indicate that 6.4 is not included in the solution set. Shade the part between these two points.

Step 6: Write in Interval Notation

The solution \(4 \leq x < 6.4\) can be written in interval notation as \([4, 6.4)\). This notation indicates that the interval includes 4 but does not include 6.4.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that show the relationship between two values or expressions. They tell us whether one side is greater than, less than, equal to, or not equal to the other. In this case, we deal with compound inequalities which involve two or more simple inequalities.To solve a compound inequality like \(3 \leq 5x - 17 < 15\), you separate it into two parts and solve each separately.

• The first inequality is \(3 \leq 5x - 17\). By performing the same operations on both sides, such as addition or subtraction followed by division or multiplication, we find that \(x \geq 4\).
• The second inequality is \(5x - 17 < 15\), solved similarly, provides \(x < 6.4\).

These steps transform the original compound inequality into a clearer range: \(4 \leq x < 6.4\).

This indicates that the solutions for \(x\) must satisfy both conditions, making these inequalities compound.

Interval Notation

Interval notation is a simple, compact way of writing subsets of real numbers. It shows the set of solutions for an inequality or a compound inequality.For any inequality, your goal is to express the variables' possible values. In our exercise, the solution from solving the inequalities is \(4 \leq x < 6.4\). We can convert this into interval notation as \([4, 6.4)\).

• The bracket \([\) means that 4 is included in the solution set, reflecting \(x \geq 4\).
• The parenthesis \()\) means that 6.4 is not included in the solution, meaning \(x < 6.4\).

Interval notation offers a tidy way to express solution sets, making communication of solutions clear and concise. Each part of this notation reflects whether endpoints are included (closed interval) or excluded (open interval).

This approach effectively summarizes the range of solutions while emphasizing the inclusivity or exclusivity of boundary points.

Graphing Inequalities

Graphing is a visual way to represent the solution set of inequalities. For the inequality \(4 \leq x < 6.4\), we'll use a number line for its representation.

Steps to Graph This Inequality:

• Start by drawing a horizontal line, which represents the set of all possible values for \(x\).
• Mark the number 4 on this line and draw a solid circle on it to note that 4 is included in the solution set: \(x \geq 4\).
• Then, find 6.4 on the line. Draw an open circle at 6.4 indicating it is not part of the solution: \(x < 6.4\).
• Shade the area between these two circles to illustrate the range \([4, 6.4)\).

Through graphing, we get an immediate visual representation of what the inequality enacts. This method provides clarity and helps confirm that the endpoints of the inequality are correctly inclusive or exclusive, complementing the analytical solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication) that make up the foundation of algebraic equations and inequalities. Understanding these is crucial for solving inequalities.In our exercise, the expression \(5x - 17\) is called an algebraic expression. It involves:

• The variable \(x\), representing unknown values.
• The constant numbers 5 and 17.
• Operations such as multiplication and subtraction.

These expressions are manipulated through various arithmetic operations to isolate the variable. By understanding and performing operations correctly, you can simplify and solve both simple and compound inequalities.

Proficiency in handling algebraic expressions is key to solving more complex mathematical problems, ensuring that solutions are both straightforward and accurate.