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Chapter 1: Problem 41

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \( x < -6\).
03

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.
04

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.

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