When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.

Verified step by step guidance

1. Let the number be represented by the variable \(x\). The expression "3 times a number" is \$3x\(, and "4 minus 3 times a number" is written as \)4 - 3x$.
2. The problem states that the absolute value of this difference is at least 5. This can be written as the inequality \(|4 - 3x| \geq 5\).
3. Recall that the absolute value inequality \(|A| \geq B\) (where \(B > 0\)) can be rewritten as two separate inequalities: \(A \leq -B\) or \(A \geq B\). Applying this, we get two inequalities: \$4 - 3x \leq -5\( or \)4 - 3x \geq 5$.
4. Solve each inequality separately for \(x\). For \$4 - 3x \leq -5\(, subtract 4 from both sides and then divide by -3, remembering to reverse the inequality sign when dividing by a negative number. For \)4 - 3x \geq 5$, do the same steps but keep the inequality direction the same.
5. Express the solutions from both inequalities in interval notation and combine them using the union symbol \(\cup\) to represent all values of \(x\) that satisfy the original absolute value inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of an algebraic expression is compared to a number. The inequality |A| ≥ B means that the expression A is either greater than or equal to B or less than or equal to -B. This concept helps in splitting the problem into two separate inequalities to solve.

Translating Word Problems into Algebraic Expressions

This involves converting verbal statements into mathematical expressions or equations. For example, '3 times a number' translates to 3x, and 'subtracted from 4' means 4 - 3x. Understanding this translation is crucial to formulating the correct inequality to solve.

Interval Notation

Interval notation is a way to represent sets of numbers on the number line. It uses parentheses and brackets to denote open or closed intervals, respectively. After solving inequalities, interval notation succinctly expresses the solution set, such as (-∞, a] ∪ [b, ∞).