Problem 115 - Algebra and Trigonometry

Question

Solve and graph the solution set on a number line: $$ \frac{2 x-3}{8} \leq \frac{3 x}{8}+\frac{1}{4} $$ (Section 1.7, Example 5)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 1\).

Step 1: Subtract \((3x)/8 \) from both sides

To isolate the variable, we'll begin by subtracting \((3x)/8\). Subtracting this expression from both sides will eliminate \(3x/8\) from the right side of the inequality and will also change the term on the left side to \((-x+3)/8\).So the inequality becomes: \((-x+3)/8 \leq 1/4 \)

Step 2: Multiply both sides by 8

To get rid of the denominator of 8 on the left, multiply both sides of the inequality by 8 to give: \(-x + 3 \leq 2\)

Step 3: Add 'x' to both sides

Next, to isolate the variable on one side, add 'x' to both sides of the inequality to give: \(3 \leq x + 2\)

Step 4: Subtract 2 from both sides

Finally, to solve for 'x', subtract 2 from both sides of the inequality to give: \(x \geq 1\)

Step 5: Graph the solution on a number-line

A number line is drawn from -2 to 3. The number '1' is then highlighted, with an arrow pointing to the right starting from 1, indicating that the solution includes all numbers greater than or equal to 1.

Key Concepts

Understanding Algebra in Inequality Solving

Algebra is the branch of mathematics that helps in finding unknown values using known quantities. When dealing with inequalities, algebra is essential as it allows us to manipulate expressions to isolate the variable. In the given exercise, we start by trying to find a relationship involving an unknown variable, \(x\).
We use operations such as addition, subtraction, multiplication, or division to simplify the inequality.
Particularly, moving terms from one side of the inequality to the other requires careful handling to maintain the inequality's nature.

• The key to solving inequalities is balancing both sides by performing the same operation.
• Unlike equations, inequalities involve signs \((\leq, \geq)\) that denote relationships other than mere equality.
• While rearranging terms, we ensure the direction of the inequality remains the same unless we multiply or divide by a negative number.

Learning algebraic manipulations like those seen in the problem enhances your problem-solving skills and your understanding of mathematical relationships.

Visualizing Solutions Using a Number Line Graph

A number line graph offers a visual representation of the solution set for inequalities. Once you solve the inequality algebraically, you can graph the resulting solution. In this exercise, the final solution is \(x \geq 1\).
Here's how the number line graph comes into play:

• Draw a horizontal line representing numbers increasing from left to right.
• Mark the significant value—in this case, '1'—on the line.
• Use a filled circle to indicate that '1' is included in the solution set, showing 'greater than or equal to'.
• Draw an arrow starting from '1' and extending to the right, symbolizing that the solution set includes all numbers greater than '1'.

The number line graph is a powerful visual aid that clarifies solutions that might look complex in algebraic form.

Mastering Variable Isolation

Variable isolation is a fundamental step in solving equations and inequalities. The main goal is to have the variable on one side and the numbers on the other. Consider the transformed inequality \(-x + 3 \leq 2\).
To isolate \(x\), you reverse operations like addition and subtraction applied to it. Some useful tips include:

• Begin by moving terms not containing \(x\) to the opposite side of the inequality.
• Perform operations inversely; if a term is subtracted, you add, and if multiplied, you divide.
• Ensure, when multiplying or dividing by a negative number, to flip the inequality sign as it impacts the relationship.

Successfully isolating the variable in the given exercise led us to the solution \(x \geq 1\), which then could be represented graphically.

Defining the Solution Set

The solution set in an inequality lists all the possible values of a variable that make the inequality true. Solving the exercise leads us to the inequality \(x \geq 1\).
This means the solution set includes all numbers that are greater than or equal to 1.
We can write this solution using set notation as \(\{x | x \geq 1\}\). Key aspects of a solution set:

• Infinite nature: In this case, all numbers starting from 1 and going to infinity satisfy the inequality.
• Inclusion: The inequality \(\geq\) includes the boundary point '1' within the solution set.
• Representation: Apart from algebraic expression, solution sets can be depicted graphically using number lines or interval notation like \([1, \infty)\).

Understanding solution sets helps you appreciate the range and scope of solutions beyond mere numerical answers.