Question

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.

Accepted Answer

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 $$4 - 3x. $$The problem states that the absolute value of this difference is at least 5. This can be written as the inequality $$|4 - 3x| \geq 5. $$Recall that the absolute value inequality $$|A| \geq B ($$where $$B \u003e 0) $$can be rewritten as two separate inequalities: $$A \leq -B or$$ $$A \geq B. $$Applying this, we get two inequalities: $$4 - 3x \leq -5 or$$ $$4 - 3x \geq 5. $$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. 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.

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.

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Key Concepts

Absolute Value Inequalities

Translating Word Problems into Algebraic Expressions

Interval Notation