Solve each inequality. Give the solution set in interval notation. | 7 - 3x | > 4

Verified step by step guidance

1. Start by considering the absolute value inequality $|7-3x|>4$. This can be split into two separate inequalities: $7-3x>4$ and $7-3x<-4$.
2. Solve the first inequality $7-3x>4$ by subtracting 7 from both sides to get $-3x>-3$. Then, divide both sides by -3, remembering to reverse the inequality sign, resulting in $x<1$.
3. Solve the second inequality $7-3x<-4$ by subtracting 7 from both sides to get $-3x<-11$. Then, divide both sides by -3, remembering to reverse the inequality sign, resulting in $x>\frac{11}{3}$.
4. Combine the solutions from both inequalities. The solution set is the union of the intervals from the two inequalities: $x<1$ and $x>\frac{11}{3}$.
5. Express the solution set in interval notation. Since the solution is the union of two intervals, it is $(-\mathrm{\infty },\frac{11}{3})\cup (1,\mathrm{\infty })$.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that measure the distance of a number from zero on the number line. The inequality |A| > B indicates that A is either greater than B or less than -B. Understanding how to break down these inequalities into two separate cases is crucial for finding the solution set.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Solving Linear Inequalities

Solving linear inequalities involves finding the values of a variable that satisfy the inequality. This process often includes isolating the variable on one side of the inequality sign and may require reversing the inequality when multiplying or dividing by a negative number. The solution is typically expressed in interval notation.