Solve each inequality. Give the solution set in interval notation. -5<5+2x<11

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Start by understanding that the inequality -5 < 5 + 2x < 11 is a compound inequality, meaning you need to solve both parts simultaneously.

First, isolate the middle expression by subtracting 5 from all three parts of the inequality: -5 - 5 < 5 + 2x - 5 < 11 - 5, which simplifies to -10 < 2x < 6.

Next, solve for x by dividing all parts of the inequality by 2 (since 2 is positive, the inequality signs remain the same): \(\frac{-10}{2}\) < \(\frac{2x}{2}\) < \(\frac{6}{2}\), which simplifies to -5 < x < 3.

Interpret the solution: x is greater than -5 and less than 3, so the solution set includes all real numbers between -5 and 3, not including -5 and 3 themselves.

Write the solution set in interval notation as ( -5, 3 ).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Compound Inequalities

A compound inequality involves two inequalities joined together, often with 'and' or 'or'. In this problem, the inequality -5 < 5 + 2x < 11 means both conditions must be true simultaneously. Solving requires isolating the variable within the combined inequality.

Solving Linear Inequalities

Solving linear inequalities involves performing algebraic operations to isolate the variable while maintaining the inequality's direction. When multiplying or dividing by a negative number, the inequality sign reverses. The goal is to find all values of x that satisfy the inequality.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate values not included (open interval), while brackets indicate inclusion (closed interval). For example, (a, b) means all numbers between a and b, excluding endpoints.