Textbook Question
Solve each inequality. Give the solution set in interval notation. 2>-6x+3>-3
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1. Equations & Inequalities
Linear Inequalities
2:19 minutesProblem 35
Textbook Question
Solve each inequality. Give the solution set in interval notation. | 5 - 3x | > 7
Verified step by step guidance1
Start by understanding that the inequality involves an absolute value: \(|5 - 3x| > 7\). Recall that \(|A| > B\) means \(A > B\) or \(A < -B\) when \(B > 0\).
Set up two separate inequalities based on the definition of absolute value:
1) \$5 - 3x > 7$
2) \$5 - 3x < -7$
Solve the first inequality \$5 - 3x > 7\(:
Subtract 5 from both sides to isolate the term with \)x\(: \)-3x > 7 - 5$
Simplify the right side: \(-3x > 2\)
Divide both sides by \(-3\), remembering to reverse the inequality sign because you are dividing by a negative number: \(x < \frac{2}{-3}\)
Solve the second inequality \$5 - 3x < -7\(:
Subtract 5 from both sides: \)-3x < -7 - 5$
Simplify the right side: \(-3x < -12\)
Divide both sides by \(-3\), reversing the inequality sign: \(x > \frac{-12}{-3}\)
Combine the two solution sets from the inequalities to express the solution in interval notation. The solution will be all \(x\) values less than \(\frac{2}{-3}\) or greater than \(\frac{-12}{-3}\).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Inequalities
Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve |A| > B, where B > 0, split the inequality into two cases: A > B or A < -B. This approach helps find all values of the variable that satisfy the inequality.
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Solving Linear Inequalities
Solving linear inequalities requires isolating the variable on one side while maintaining the inequality's direction. When multiplying or dividing by a negative number, the inequality sign must be reversed. This process yields the range of values that satisfy the inequality.
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Interval Notation
Interval notation is a concise way to represent solution sets of inequalities using intervals. Parentheses () denote values not included (open intervals), while brackets [] denote included values (closed intervals). It clearly shows the range of solutions on the number line.
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