Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(-4<\frac{2 x-3}{3}<4$$-4<\frac{2 x-3}{3}<4\)

Step-by-Step Solution

Answer

The solution to the inequality is \(-4.5

Step 1: Solving the first inequality

To solve \(-4<\frac{2 x-3}{3}\), multiply all parts by 3 to clear the denominator, resulting in \(-12<2x-3\). Then, add 3 to all parts to isolate \(x\) on one side, yielding \(-9<2x\). Finally, divide all parts by 2 to solve for \(x\), which gives \(-4.5-4.5\).

Step 2: Solving the second inequality

To solve \(\frac{2 x-3}{3}<4\), multiply all parts by 3 to clear the denominator, resulting in \(2x-3<12\). Then, add 3 to all parts to isolate \(x\) on one side, yielding \(2x<15\). Finally, divide all parts by 2 to solve for \(x\), which gives \(x<7.5\).

Step 3: Finding the Intersection

To find the intersection of the solution sets, combine the solution sets of both inequalities. This gives you \(-4.5

Step 4: Graphing the solution

On the number line, mark two points, -4.5 and 7.5. Since there are no 'or equal to' conditions, both points will be open (unfilled). Draw a line connecting these points. This is the graph of the solution set for this inequality.

Key Concepts

Graphing Solutions

Graphing solutions of inequalities is a visual way to display which values satisfy the inequality. It helps make the solution more intuitive and understandable. When graphing, we use a number line to represent possible solutions. For open intervals (when the inequality is strict, like < or >), we use open circles. Here's how to graph our example:

• Identify the interval from the solution, here it's \(-4.5 < x < 7.5\).
• On the real number line, place open circles at -4.5 and 7.5 to indicate these values are not included in the solution.
• Draw a line between these points to represent all numbers between them that satisfy the inequality.

Graphing provides a clear visual of where the solution set lies on the real number line.

Real Number Line

The real number line is a straight line that represents all real numbers. It's a fundamental concept in mathematics, allowing us to visualize numbers and their relationships effectively. Every point on the line corresponds to a real number, and distances are meaningful.
Here are some key points about the real number line:

• The number line extends infinitely in both directions.
• Zero typically occupies the middle, with positive numbers to the right and negative numbers to the left.
• It's particularly useful in graphing inequalities, as it provides a one-dimensional space to mount points and intervals.

Understanding the real number line is crucial for comprehending intervals and solution sets in algebra.

Algebraic Manipulation

Algebraic manipulation involves a variety of techniques used to solve equations and inequalities. These include simplifying expressions, clearing fractions, and isolating variables. Such operations are essential for solving problems like our inequality.
Here's how you can carry out algebraic manipulations effectively:

• Start by clearing fractions or decimals by multiplying through by a factor, which simplifies the equation.
• Use addition or subtraction to move terms to one side of the equation, helping to isolate the variable.
• Finally, divide or multiply to solve for the variable completely.

These steps require careful attention to ensure the operations are valid, particularly when dealing with inequalities where multiplying or dividing by a negative number reverses the inequality sign. By mastering these manipulations, solving complex inequalities becomes much simpler.

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Key Concepts

Graphing Solutions

Graphing solutions of inequalities is a visual way to display which values satisfy the inequality. It helps make the solution more intuitive and understandable. When graphing, we use a number line to represent possible solutions. For open intervals (when the inequality is strict, like < or >), we use open circles. Here's how to graph our example:

• Identify the interval from the solution, here it's \(-4.5 < x < 7.5\).
• On the real number line, place open circles at -4.5 and 7.5 to indicate these values are not included in the solution.
• Draw a line between these points to represent all numbers between them that satisfy the inequality.

Graphing provides a clear visual of where the solution set lies on the real number line.

Real Number Line

The real number line is a straight line that represents all real numbers. It's a fundamental concept in mathematics, allowing us to visualize numbers and their relationships effectively. Every point on the line corresponds to a real number, and distances are meaningful.
Here are some key points about the real number line:

• The number line extends infinitely in both directions.
• Zero typically occupies the middle, with positive numbers to the right and negative numbers to the left.
• It's particularly useful in graphing inequalities, as it provides a one-dimensional space to mount points and intervals.

Understanding the real number line is crucial for comprehending intervals and solution sets in algebra.

Algebraic Manipulation

Algebraic manipulation involves a variety of techniques used to solve equations and inequalities. These include simplifying expressions, clearing fractions, and isolating variables. Such operations are essential for solving problems like our inequality.
Here's how you can carry out algebraic manipulations effectively:

• Start by clearing fractions or decimals by multiplying through by a factor, which simplifies the equation.
• Use addition or subtraction to move terms to one side of the equation, helping to isolate the variable.
• Finally, divide or multiply to solve for the variable completely.

These steps require careful attention to ensure the operations are valid, particularly when dealing with inequalities where multiplying or dividing by a negative number reverses the inequality sign. By mastering these manipulations, solving complex inequalities becomes much simpler.