Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x - 2 ≤ 1 + x

Verified step by step guidance

1. Start by isolating the variable term on one side of the inequality. Add \(2x\) to both sides to move the \(-2x\) term from the left to the right side: \(-2x - 2 + 2x \leq 1 + x + 2x\).
2. Simplify both sides: the left side becomes \(-2\), and the right side becomes \(1 + 3x\), so the inequality is now \(-2 \leq 1 + 3x\).
3. Next, isolate the term with \(x\) by subtracting \(1\) from both sides: \(-2 - 1 \leq 1 + 3x - 1\), which simplifies to \(-3 \leq 3x\).
4. To solve for \(x\), divide both sides of the inequality by \(3\). Since \(3\) is positive, the inequality direction remains the same: \(\frac{-3}{3} \leq \frac{3x}{3}\), which simplifies to \(-1 \leq x\).
5. Express the solution set in interval notation. Since \(x\) is greater than or equal to \(-1\), the solution set is \([ -1, \infty )\).

Key Concepts

Solving Linear Inequalities

Linear inequalities involve expressions with variables raised to the first power and inequality signs. To solve them, isolate the variable on one side by performing algebraic operations like addition, subtraction, multiplication, or division, while carefully reversing the inequality sign when multiplying or dividing by a negative number.

Properties of Inequalities

Inequalities follow specific rules: adding or subtracting the same number on both sides keeps the inequality direction unchanged, but multiplying or dividing both sides by a negative number reverses the inequality sign. Understanding these properties is essential to correctly manipulate and solve inequalities.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities. It uses parentheses () for values not included (open intervals) and brackets [] for values included (closed intervals), indicating the range of values that satisfy the inequality.