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)/6 ≥ (x - 2)/9 + 5/18

Verified step by step guidance

1. Start by writing down the inequality: \(\frac{(x - 4)}{6} \geq \frac{(x - 2)}{9} + \frac{5}{18}\).
2. To eliminate the fractions, find the least common denominator (LCD) of 6, 9, and 18, which is 18. Multiply every term on both sides of the inequality by 18 to clear the denominators.
3. After multiplying, simplify each term: \(18 \times \frac{(x - 4)}{6} = 3(x - 4)\), \(18 \times \frac{(x - 2)}{9} = 2(x - 2)\), and \(18 \times \frac{5}{18} = 5\).
4. Rewrite the inequality without fractions: \(3(x - 4) \geq 2(x - 2) + 5\). Then, distribute the constants inside the parentheses: \(3x - 12 \geq 2x - 4 + 5\).
5. Combine like terms on the right side: \$2x - 4 + 5 = 2x + 1\(. Now, solve the inequality \(3x - 12 \geq 2x + 1\) by isolating \)x$ on one side.

Key Concepts

Solving Linear Inequalities

Linear inequalities involve expressions with variables raised to the first power and inequality symbols (>, <, ≥, ≤). Solving them requires isolating the variable on one side while maintaining the inequality's direction, often by performing algebraic operations like addition, subtraction, multiplication, or division.

Manipulating Fractions in Inequalities

When solving inequalities with fractions, it is important to find a common denominator or clear fractions by multiplying both sides by the least common denominator. Care must be taken when multiplying or dividing by negative numbers, as this reverses the inequality sign.

Interval Notation and Graphing Solutions

Interval notation expresses solution sets as intervals on the number line, using parentheses for strict inequalities and brackets for inclusive inequalities. Graphing these solutions involves shading the appropriate region on a number line and indicating boundary points as open or closed dots based on the inequality.