Problem 41 - College Algebra

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(1-\frac{x}{2}>4\)

Step-by-Step Solution

Answer

The solution to the inequality \(1-\frac{x}{2}>4\) is \( x < -6\), which is expressed in interval notation as \((- \infty, -6)\). On a number line, the solution is all points to the left of -6, with -6 not included in the solution set.

Step 1: Simplify the Inequality

To simplify this inequality, it can be helpful to first eliminate the fraction. Multiply both sides by 2, giving \(2 \cdot (1 - \frac{x}{2}) > 2 \cdot 4\), which simplifies to \(2 - x > 8\). This equivalent inequality is easier to solve.

Step 2: Solve for x

Next, subtract 2 from both sides to isolate \(x\) in the inequality, producing \(-x > 6\). For a more familiar format, divide everything by -1, being sure to reverse the direction of the inequality as is required when multiplying or dividing an inequality by a negative number. This results in \( x < -6\).

Step 3: Convert Into Interval Notation

The inequality \( x < -6\) can be expressed in interval notation as \((- \infty, -6)\). This represents all real numbers less than -6.

Step 4: Graph on a number line

On the number line, the solution is represented as all points to the left of -6. A parenthesis is placed at -6 to indicate that -6 is not included in the solution set.