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 with the given inequality: \$1 - \frac{x}{2} > 4$.
2. Isolate the term containing \(x\) by subtracting 1 from both sides: \$1 - \frac{x}{2} - 1 > 4 - 1\(, which simplifies to \)- \frac{x}{2} > 3$.
3. To solve for \(x\), multiply both sides of the inequality by \(-2\) to eliminate the fraction. Remember, when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign: \(x < -6\).
4. Express the solution in interval notation. Since \(x\) is less than \(-6\), the solution set is \((-\infty, -6)\).
5. To graph the solution on a number line, draw a number line, mark the point \(-6\) with an open circle (because \(x\) is strictly less than \(-6\), not equal), and shade all values to the left of \(-6\).

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 inverse operations, similar to solving linear equations, but pay attention to inequality rules.

Effect of Multiplying or Dividing by Negative Numbers

When solving inequalities, multiplying or dividing both sides by a negative number reverses the inequality sign. This rule is crucial to maintain the inequality's truth and must be applied carefully during the solution process.

Interval Notation and Number Line Graphing

Interval notation expresses solution sets using parentheses and brackets to indicate open or closed intervals. Graphing on a number line visually represents these solutions, showing which values satisfy the inequality and whether endpoints are included.