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/2 > 4

Verified step by step guidance

1. Start by isolating the variable term on one side of the inequality. Subtract 1 from both sides: $-\frac{x}{2}>4-1$.
2. Simplify the right side of the inequality: $-\frac{x}{2}>3$.
3. To eliminate the fraction, multiply both sides of the inequality by -2. Remember to reverse the inequality sign when multiplying by a negative number: $x<-6$.
4. Express the solution in interval notation. Since $x<-6$, the interval is $(-\mathrm{\infty },-6)$.
5. Graph the solution set on a number line. Draw an open circle at -6 and shade the line to the left, indicating all numbers less than -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 all numbers greater than 2 and up to 5, including 5 but not 2. This notation is particularly useful for expressing solution sets of inequalities.

Graphing on a Number Line

Graphing on a number line involves visually representing the solution set of an inequality. Each point on the number line corresponds to a real number, and the solution set is indicated by shading the appropriate region. Open circles are used for endpoints that are not included in the solution, while closed circles indicate included endpoints. This visual representation helps in understanding the range of solutions for the inequality.