Problem 70 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{1}{4} x-\frac{1}{3} \leq x+2 $$

Step-by-Step Solution

Answer

The solution is \([-\frac{28}{9}, \infty)\).

Step 1: Isolate the Variable Terms

First, bring all terms including the variable to one side of the inequality. Start by subtracting \(x\) from both sides of the inequality:\[\frac{1}{4}x - \frac{1}{3} - x \leq x + 2 - x\]This simplifies to:\[\frac{1}{4}x - x - \frac{1}{3} \leq 2\]

Step 2: Combine Like Terms

Now, combine the \(x\) terms on the left-hand side:\[\left(\frac{1}{4} - 1\right)x - \frac{1}{3} \leq 2\]\[-\frac{3}{4}x - \frac{1}{3} \leq 2\]

Step 3: Eliminate the Constant Term

Add \(\frac{1}{3}\) to both sides to isolate the \(x\) term:\[-\frac{3}{4}x - \frac{1}{3} + \frac{1}{3} \leq 2 + \frac{1}{3}\]Simplify the right-hand side:\[-\frac{3}{4}x \leq \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\]

Step 4: Solve for x

Now, divide both sides by \(-\frac{3}{4}\) to solve for \(x\). Remember that dividing by a negative number reverses the inequality:\[x \geq \frac{7}{3} \times \frac{-4}{3} = \frac{-28}{9}\]

Step 5: Graph the Solution Set

On a number line, plot \(\frac{-28}{9}\) and shade everything to the right. Use a closed circle at \(\frac{-28}{9}\) to indicate that it is included in the solution set.

Step 6: Write the Solution in Interval Notation

The solution in interval notation includes all numbers greater than or equal to \(-\frac{28}{9}\). This is written as:\[\left[-\frac{28}{9}, \infty\right)\]

Key Concepts

Interval Notation

When solving inequalities, expressing the solution in interval notation offers a compact and clear format. Interval notation encapsulates the set of solutions using brackets and parentheses to convey the limits of the inequality.

For example, a solution involving no upper limit uses the infinity symbol \(\infty\). An interval might look like \([-28/9, \infty)\), indicating that \(-28/9\) is included and the set extends infinitely to the right.

Here's how to understand this notation:

• Closed bracket [ ] - Includes the endpoint within the solution set.
• Open parenthesis ( ) - Excludes the endpoint from the solution set.

Remember in our exercise, the solution \([-28/9, \infty)\) means all values of \(x\) starting from \(-28/9\) and going upwards are solutions, including \(-28/9\) itself.

Graphing Inequalities

Graphing inequalities helps visualize the range of possible solutions. It offers a graphical interpretation of the set of numbers included in the solution.

To graph the inequality \(x \geq -\frac{28}{9}\), follow these steps:

• Draw a number line and locate the point \(-\frac{28}{9}\).
• Place a closed circle over \(-\frac{28}{9}\) to show it is part of the solution set.
• Shade the region to the right of \(-\frac{28}{9}\), extending infinitely, to include all larger values.

Using these steps, the graphical representation corresponds to the interval \([-\frac{28}{9}, \infty)\). Graphing not only confirms our interval notation but solidifies our understanding of all possible solutions for the inequality.

Isolation of Variables

In the context of solving inequalities, the isolation of variables is a fundamental step. This process involves manipulating the inequality to get the variable, often \(x\), alone on one side.

Let's break down how to achieve this:

• Move variables: Use operations like addition or subtraction to relocate \(x\) terms to one side.
• Combine like terms: Simplify the expression by merging similar terms.
• Solve for the variable: Divide or multiply the entire inequality to isolate \(x\). Recall that dividing or multiplying by a negative number requires reversing the inequality sign.

In our example, isolating \(x\) involved moving terms, simplifying, and then dividing by a negative fraction, which reversed the inequality symbol to yield the final result \(x \geq -\frac{28}{9}\). This systematic approach ensures clarity and accuracy, helping you master solving inequalities.