In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. x/4 - 3/2 ≤ x/2 + 1

Verified step by step guidance

1. Start by writing down the inequality: \(\frac{x}{4} - \frac{3}{2} \leq \frac{x}{2} + 1\).
2. To eliminate the fractions, find the least common denominator (LCD), which is 4, and multiply every term on both sides of the inequality by 4: \(4 \times \left(\frac{x}{4} - \frac{3}{2}\right) \leq 4 \times \left(\frac{x}{2} + 1\right)\).
3. Distribute the 4 to each term inside the parentheses: \(x - 4 \times \frac{3}{2} \leq 4 \times \frac{x}{2} + 4 \times 1\).
4. Simplify each term: \(x - 6 \leq 2x + 4\).
5. Next, isolate the variable terms on one side and constants on the other by subtracting \(x\) from both sides and subtracting 4 from both sides: \(x - x - 6 - 4 \leq 2x - x + 4 - 4\), which simplifies to \(-10 \leq x\).

Key Concepts

Linear Inequalities

Linear inequalities involve expressions with variables raised to the first power and inequality symbols like ≤, <, ≥, or >. Solving them requires isolating the variable on one side while maintaining the inequality's truth, often by performing algebraic operations similar to those used in equations.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. It uses parentheses () for values not included and brackets [] for values included, such as [a, b] meaning all numbers from a to b including both endpoints. This notation is essential for expressing solution sets of inequalities.

Graphing on a Number Line

Graphing solutions on a number line visually represents the set of values that satisfy an inequality. Points or intervals are marked with open circles for excluded endpoints and closed circles for included endpoints, helping to clearly communicate the solution set's range.