Problem 19 - Calculus Single Variable

Question

In Exercises \(19-24,\) sketch the set on a real number line. \(\\{x: 2 x-5

Step-by-Step Solution

Answer

The inequality \(x < 9\) on a number line is shaded left from an open circle at 9.

Step 1: Simplify the Inequality

Start by simplifying the inequality: \(2x - 5 < x + 4\). To make it easier to solve, we will first subtract \(x\) from both sides of the inequality. This gives us \(2x - x - 5 < x - x + 4\), which simplifies to \(x - 5 < 4\).

Step 2: Solve for x

Next, solve the simpler inequality \(x - 5 < 4\). Add 5 to both sides to isolate \(x\): \(x - 5 + 5 < 4 + 5\). This simplifies to \(x < 9\).

Step 3: Interpret the Inequality

The solution \(x < 9\) requires that \(x\) can be any real number less than 9. On the number line, this is represented by an open circle at 9 and shading to the left, which includes all real numbers less than 9.

Key Concepts

Inequalities

Inequalities are a fundamental concept that express the relationship between two values using symbols like "<" (less than), ">" (greater than), "≤" (less than or equal to), and "≥" (greater than or equal to).
They play a central role in algebra and are used to describe ranges of values. Here, the inequality given is \(2x - 5 < x + 4\).
To solve such inequalities, you aim to find all possible values of the variable that satisfy the condition.Applying operations such as addition, subtraction, multiplication, or division can help simplify and eventually solve the inequality. For example, subtracting \(x\) from both sides of \(2x - 5 < x + 4\) simplifies the expression, making it easier to solve.
Like equations, inequalities can have infinite solutions, and the solution is often expressed as a range or set on the number line.

Graphical Representation

Graphical representation of inequalities is crucial in visualizing solutions. For the inequality \(x < 9\), graphing determines the span of values it denotes.
Unlike equations, where you find fixed solutions, inequalities describe a series of values.In this case, the graphical representation involves:

• Placing an open circle at \(9\) on the real number line.
• The open circle indicates that \(9\) itself is not included in the solution set.
• Shading continuously to the left of \(9\), indicates all numbers less than \(9\) satisfy the inequality.

The graphical method aids in understanding the range of solutions not merely as numbers but as a part of a continuum, which is especially helpful in analyzing compound inequalities.

Number Line Visualization

Number lines are a simple yet powerful tool for visualizing real numbers and solutions to inequalities. They span horizontally and represent every real number along a line.
In representing \(x < 9\) on a number line:

• Mark a point on the line at \(9\).
• Use an open circle if the boundary value isn't part of the solution, like in our case with \(9\).
• Shaded areas on the number line illustrate the range of values that satisfy the inequality.

Through number line visualization, complex algebraic solutions can become more relatable and transparent, making it easier to grasp the breadth of possible solutions and see how they relate to one another.

Algebraic Manipulation

Algebraic manipulation involves changing an expression or equation to simplify or find solutions. It includes operations such as adding, subtracting, multiplying, and dividing both sides of an equation or inequality by a constant value.
The key steps in manipulating the initial inequality, \(2x - 5 < x + 4\), include:

• Eliminating the term \(x\) on the right by subtracting \(x\) from both sides, simplifying the inequality to \(x - 5 < 4\).
• Isolating \(x\) by adding \(5\) to both sides, leading us to the solution \(x < 9\).

Manipulation helps reveal the core relationship between variables in an inequality.
It's crucial to perform these steps carefully, especially when multiplying or dividing by negative numbers, as it would require reversing the inequality symbol. Such careful algebraic manipulation ultimately provides clear solutions that can be graphically represented.