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 2 to both sides: \(-2x - 2 + 2 \leq 1 + x + 2\).
2. Simplify the inequality: \(-2x \leq 3 + x\).
3. Next, eliminate the \(x\) on the right side by subtracting \(x\) from both sides: \(-2x - x \leq 3 + x - x\).
4. Simplify again: \(-3x \leq 3\).
5. Finally, solve for \(x\) by dividing both sides by \(-3\). Remember to reverse the inequality sign when dividing by a negative number: \(x \geq -1\).

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They use symbols such as '≤' (less than or equal to) and '≥' (greater than or equal to) to indicate the direction of the relationship. Solving inequalities involves finding the values of the variable that make the inequality true, which can often lead to a range of solutions rather than a single value.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses brackets '[ ]' to include endpoints and parentheses '( )' to exclude them. For example, the interval [1, 5) includes 1 and all numbers up to but not including 5. This notation is particularly useful for expressing the solution sets of inequalities succinctly.

Solving Linear Inequalities

Solving linear inequalities involves manipulating the inequality in a manner similar to solving linear equations, but with special attention to the direction of the inequality sign. When multiplying or dividing by a negative number, the inequality sign must be reversed. The goal is to isolate the variable on one side, allowing for the determination of the solution set, which can then be expressed in interval notation.