Solve each inequality. Give the solution set using interval notation. See Example 10. -5 < 5 + 2x < 11

검증된 단계별 안내

Start by understanding that the compound inequality \(-5 < 5 + 2x < 11\) means both inequalities \(-5 < 5 + 2x\) and \(5 + 2x < 11\) must be true simultaneously.

Isolate the variable \(x\) in the first inequality: subtract 5 from all parts to get \(-5 - 5 < 2x\), which simplifies to \(-10 < 2x\).

Next, isolate \(x\) in the second inequality: subtract 5 from all parts to get \(2x < 11 - 5\), which simplifies to \(2x < 6\).

Now, solve for \(x\) in both inequalities by dividing all parts by 2 (note that dividing by a positive number does not change the inequality direction): from \(-10 < 2x\) we get \(-5 < x\), and from \(2x < 6\) we get \(x < 3\).

Combine the two results to write the solution set as \(-5 < x < 3\), which in interval notation is expressed 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, such as -5 < 5 + 2x < 11. To solve it, you treat it as two separate inequalities and find the values of the variable that satisfy both simultaneously.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side by performing inverse operations, similar to solving equations, but remembering to reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation

Interval notation is a way to express the solution set of inequalities using intervals. It uses parentheses for values not included and brackets for values included, clearly showing the range of possible solutions.