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. 1 - (x + 3) ≥ 4 - 2x

Verified step by step guidance

1. Distribute the negative sign on the left side: 1 - x - 3 \geq 4 - 2x.
2. Combine like terms on the left side: -x - 2 \geq 4 - 2x.
3. Add 2x to both sides to get all x terms on one side: -x + 2x - 2 \geq 4.
4. Simplify the left side: x - 2 \geq 4.
5. Add 2 to both sides to isolate x: x \geq 6.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that involve a linear function and an inequality sign (such as <, >, ≤, or ≥). They represent a range of values rather than a single solution. Solving a linear inequality involves isolating the variable on one side of the inequality, similar to solving an equation, but requires special attention to the direction of the inequality when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval, [ ]) or excluded (open interval, ( )). For example, the interval [2, 5) includes 2 but excludes 5, representing all numbers between 2 and 5, including 2 but not 5.

Graphing on a Number Line

Graphing solution sets on a number line visually represents the range of values that satisfy an inequality. Closed circles indicate that an endpoint is included in the solution, while open circles indicate it is not. The line or shading between the points shows all the values that are part of the solution set, helping to illustrate the concept of intervals and the solutions to inequalities.