Textbook Question
Solve each inequality. Give the solution set using interval notation. 5 ≤ 2x -3 ≤ 7
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1. Equations & Inequalities
Linear Inequalities
5:04 minutesProblem 91
Textbook Question
In Exercises 59–94, solve each absolute value inequality.12 < |- 2x + 6/7| + 3/7
Verified step by step guidance1
Start by isolating the absolute value expression. Subtract \( \frac{3}{7} \) from both sides of the inequality: \( 12 - \frac{3}{7} < \left| -2x + \frac{6}{7} \right| \).
Simplify the left side of the inequality: \( \frac{84}{7} - \frac{3}{7} = \frac{81}{7} \), so the inequality becomes \( \frac{81}{7} < \left| -2x + \frac{6}{7} \right| \).
Recognize that the absolute value inequality \( \left| A \right| > B \) can be rewritten as two separate inequalities: \( A > B \) or \( A < -B \).
Apply this to the expression: \( -2x + \frac{6}{7} > \frac{81}{7} \) or \( -2x + \frac{6}{7} < -\frac{81}{7} \).
Solve each inequality separately for \( x \). For the first inequality, subtract \( \frac{6}{7} \) from both sides and then divide by \(-2\). For the second inequality, subtract \( \frac{6}{7} \) from both sides and then divide by \(-2\), remembering to reverse the inequality sign when dividing by a negative number.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value
Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. In the context of inequalities, understanding how to manipulate absolute values is crucial for solving equations and inequalities that involve them.
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Inequalities
Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving absolute value inequalities, it is important to consider both the positive and negative scenarios that arise from the definition of absolute value, leading to two separate cases to solve.
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Isolating the Absolute Value
To solve an absolute value inequality, the first step is often to isolate the absolute value expression on one side of the inequality. This involves performing algebraic operations such as addition, subtraction, multiplication, or division. Once isolated, the inequality can be split into two separate inequalities, allowing for a systematic approach to find the solution set.
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