Problem 111 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq -5\), which is represented by shading all values to the left of -5, including -5, on a number line.

Step 1: Clear the Fractions

Multiply each term in the inequality by 4 to cancel the denominators. This results in \(2x - 3 \geq 3x + 2\).

Step 2: Rearranging the Inequality

To isolate the variable, subtract 2x from both sides of the inequality to get \(-3 \geq x + 2\). Then subtract 2 from both sides to get \(-5 \geq x \) or \(x \leq -5\).

Step 3: Graph the Solution

On the number line, put a closed circle at -5 (because the inequality symbol is \(\leq\), so -5 is included in the solution), and shade in the area to the left of -5, as this area represents all the possible solutions.

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial step in solving and visualizing solutions. When you have an inequality like \(x \leq -5\), the goal is to represent all possible values of \(x\) that make the inequality true on a graph. To do this:

• Identify if the inequality includes the number itself (using symbols like \(\leq\) or \(\geq\)) or if it is just less or greater excluding the number (using \( < \) or \( > \)).
• Draw a number line, which is a simple horizontal line labeled with numbers.
• Place a closed circle on the number, such as -5, if the inequality includes the number itself.
• Shade the entire region on the number line that satisfies the inequality. For \(x \leq -5\), this means shading everything to the left of -5.

Graphing inequalities helps you visually see all the possible solutions in a clear and straightforward way.

Clearing Fractions

Fractions in inequalities can make them appear more complicated, but clearing fractions simplifies the problem. This involves eliminating all fractions by multiplying each term by a common denominator.

• In the example, the common denominator was 4. By multiplying every term by 4, all fractions were removed, transforming the inequality, \(\frac{2x-3}{4} \geq \frac{3x}{4} + \frac{1}{2}\), into a simpler form: \(2x - 3 \geq 3x + 2\).

This process simplifies the path to solving the inequality and ensures that you can work with whole numbers instead, which are generally easier to manage.
In every equation dealing with fractions, it is important to find the least common denominator first to apply this tactic seamlessly.
Once the fractions are cleared, you can move forward to isolate the variable and solve the inequality more efficiently.

Variable Isolation

Isolating the variable is the next crucial step after clearing fractions. This means rearranging the inequality so that the variable you are solving for is by itself on one side of the inequality.

• In the example provided, after clearing fractions, you have \(2x - 3 \geq 3x + 2\). Begin by eliminating one of the variable terms. Here, subtract \(2x\) from both sides to focus on isolating \(x\), resulting in \(-3 \geq x + 2\).
• Next, complete the isolation by removing any constants from the variable side. Subtract 2 from each side to solve for \(x\), yielding \(-5 \geq x\) or equivalently \(x \leq -5\).

Through these steps, every action performed is systematically undoing the operations surrounding the variable, bringing it into focus on its own.

Number Line Graphing

Graphing on a number line is a straightforward approach to visually present inequalities. Here's how you can execute it effectively:

• Draw a horizontal line (number line).
• Mark points on the number line including the specific value from the inequality (e.g., -5 in \(x \leq -5\)).
• For inequalities such as \(x \leq -5\), put a closed circle at -5 to show that it's included in the solutions, as indicated by the \(\leq\) symbol.
• Shade or draw a line extending from this point to indicate all numbers that satisfy the inequality. For \(x \leq -5\), shade to the left of -5 because these represent all the possible values for \(x\).

This visual representation aids in better understanding and verifying the solutions that meet the inequality conditions.