Solve each inequality. Give the solution set in interval notation. 2>-6x+3>-3

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Start by understanding the compound inequality: \(2 > -6x + 3 > -3\). This means that \(-6x + 3\) is simultaneously less than 2 and greater than -3.

Break the compound inequality into two separate inequalities: 1) \(2 > -6x + 3\) 2) \(-6x + 3 > -3\)

Solve the first inequality \(2 > -6x + 3\) by isolating \(x\): - Subtract 3 from both sides: \(2 - 3 > -6x\) - Simplify: \(-1 > -6x\) - Divide both sides by \(-6\) (remember to reverse the inequality sign when dividing by a negative number): \(\frac{-1}{-6} < x\)

Solve the second inequality \(-6x + 3 > -3\) by isolating \(x\): - Subtract 3 from both sides: \(-6x > -3 - 3\) - Simplify: \(-6x > -6\) - Divide both sides by \(-6\) (again, reverse the inequality sign): \(x < \frac{-6}{-6}\)

Combine the two inequalities to find the solution set for \(x\): \(\frac{-1}{-6} < x < \frac{-6}{-6}\) Simplify the fractions and express the solution in interval notation.

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주요 개념

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Compound Inequalities

Compound inequalities involve two inequalities joined together, often with 'and' or 'or'. In this problem, the compound inequality 2 > -6x + 3 > -3 means both inequalities must be true simultaneously. Solving requires splitting it into two separate inequalities and finding the intersection of their solution sets.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side by performing inverse operations, similar to solving equations. When multiplying or dividing by a negative number, the inequality sign must be reversed. This ensures the solution set correctly reflects the inequality's direction.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate that endpoints are not included, while brackets mean they are included. It provides a clear and standardized method to express ranges of values satisfying the inequality.