Solve each inequality. Give the solution set in interval notation. -(2/3)x-(1/6)x+(2/3)(x+1)≤4/3

Verified step by step guidance

1. First, rewrite the inequality clearly: \(-\frac{2}{3}x - \frac{1}{6}x + \frac{2}{3}(x + 1) \leq \frac{4}{3}\).
2. Distribute the \(\frac{2}{3}\) across the terms inside the parentheses: \(\frac{2}{3} \times x\) and \(\frac{2}{3} \times 1\) to get \(\frac{2}{3}x + \frac{2}{3}\).
3. Combine like terms involving \(x\): \(-\frac{2}{3}x - \frac{1}{6}x + \frac{2}{3}x\) by finding a common denominator and adding the coefficients.
4. After combining the \(x\) terms, combine the constant terms on the left side and then isolate the variable term on one side by subtracting or adding constants as needed.
5. Finally, solve for \(x\) by dividing both sides by the coefficient of \(x\), and express the solution set in interval notation, remembering to consider the direction of the inequality.

Key Concepts

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This simplifies expressions and makes solving inequalities easier. For example, terms with x can be combined by adding their coefficients.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable on one side while maintaining the inequality's direction. Operations like addition, subtraction, multiplication, or division are applied, but multiplying or dividing by a negative number reverses the inequality sign.

Interval Notation

Interval notation expresses the solution set of inequalities using intervals on the number line. Parentheses indicate open intervals (excluding endpoints), and brackets indicate closed intervals (including endpoints). It provides a concise way to represent all possible solutions.