Problem 92 - College Algebra

Question

Solve absolute value inequality. \(1<\left|x-\frac{11}{3}\right|+\frac{7}{3}\)

Step-by-Step Solution

Answer

The solution to the given inequality is \(2\frac{1}{3}

Step 1: Isolate the Absolute Value

Subtract \(\frac{7}{3}\) from both sides of the inequality: \(1-\frac{7}{3}<\left|x-\frac{11}{3}\right|\). This simplifies into \(-\frac{4}{3}<\left|x-\frac{11}{3}\right|\).

Step 2: Convert to Compound Inequality

The inequality \(-\frac{4}{3}< |x-\frac{11}{3}|\) will be converted into \(-\frac{4}{3}

Step 3: Solve the Compound Inequality

To solve the inequality \(-\frac{4}{3}

Key Concepts

Compound Inequality

Understanding compound inequalities is crucial when tackling complex problem sets involving multiple comparisons. A compound inequality contains at least two distinct inequalities joined by 'and' (intersection) or 'or' (union). An example of a compound inequality would be \( 1 < x < 3 \) which means x is greater than 1 and less than 3 simultaneously.

When solving absolute value inequalities, such as \( 1 < \left| x - \frac{11}{3} \right| + \frac{7}{3} \), we often convert them into a compound inequality. The conversion process involves recognizing that the absolute value expression describes a range of values and setting up two separate inequalities, both of which must be true. This step is essential as it simplifies the complex absolute value inequality into a more manageable form, allowing for straightforward resolutions to each part of the compound inequality.

Isolate Absolute Value

When facing an absolute value inequality like \( 1 < \left| x - \frac{11}{3} \right| + \frac{7}{3} \), the initial step is to isolate the absolute value expression. This involves moving all other terms to the other side of the inequality.

In our exercise, subtracting \( \frac{7}{3} \) from both sides \( 1 - \frac{7}{3} < \left| x - \frac{11}{3} \right| \) achieves this isolation, leading to \( -\frac{4}{3} < \left| x - \frac{11}{3} \right| \). Isolation is a key tactic as it allows us to focus directly on the properties of the absolute value, setting the stage for the next move—creating a compound inequality.

Solving Inequalities

Solving inequalities is similar to solving equations, but we must remember that the direction of the inequality changes when we multiply or divide by a negative number. Our task does not require this step; instead, we simply add or subtract the same value across the inequality to maintain the relationship.

In our exercise, we added \( \frac{11}{3} \) to each part of the compound inequality resulting from the absolute value to find the solution. This action preserves the direction of the inequality and results in \( \frac{7}{3} < x < \frac{15}{3} \), which further simplifies down to \( 2\frac{1}{3} < x < 5 \). The important takeaway is the systematic approach to such inequalities—manipulating them step by step while cautiously observing the signs and maintaining the balance across the inequality.

Absolute Value Properties

The absolute value of a number refers to its distance from zero on the number line, disregarding its sign. This is why \( |x| \) is always non-negative. Two key properties to remember are: \( |a| = a \) if \( a \) is non-negative, and \( |a| = -a \) if \( a \) is negative. Additionally, for equations involving absolute values, \( |a| = b \) means \( a \) could be \( b \) or \( -b \) because both would have the same absolute value.

In the context of inequalities like \( |x - a| < b \) we interpret this as \( x \) lying within a distance \( b \) from \( a \) in both the positive and negative directions. This is why it breaks down into the compound inequality \( -b < x - a < b \) during the solving process, implicitly acknowledging that the variable \( x \) could be \( a + b \) or \( a - b \) away from \( a \) on the number line. Understanding these properties makes it easier to interpret and solve absolute value inequalities.

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Problem 92

Other exercises in this chapter

Problem 91

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Problem 91

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Problem 92

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Problem 92

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Key Concepts

Compound Inequality

Understanding compound inequalities is crucial when tackling complex problem sets involving multiple comparisons. A compound inequality contains at least two distinct inequalities joined by 'and' (intersection) or 'or' (union). An example of a compound inequality would be \( 1 < x < 3 \) which means x is greater than 1 and less than 3 simultaneously.

When solving absolute value inequalities, such as \( 1 < \left| x - \frac{11}{3} \right| + \frac{7}{3} \), we often convert them into a compound inequality. The conversion process involves recognizing that the absolute value expression describes a range of values and setting up two separate inequalities, both of which must be true. This step is essential as it simplifies the complex absolute value inequality into a more manageable form, allowing for straightforward resolutions to each part of the compound inequality.

Isolate Absolute Value

When facing an absolute value inequality like \( 1 < \left| x - \frac{11}{3} \right| + \frac{7}{3} \), the initial step is to isolate the absolute value expression. This involves moving all other terms to the other side of the inequality.

In our exercise, subtracting \( \frac{7}{3} \) from both sides \( 1 - \frac{7}{3} < \left| x - \frac{11}{3} \right| \) achieves this isolation, leading to \( -\frac{4}{3} < \left| x - \frac{11}{3} \right| \). Isolation is a key tactic as it allows us to focus directly on the properties of the absolute value, setting the stage for the next move—creating a compound inequality.

Solving Inequalities

Solving inequalities is similar to solving equations, but we must remember that the direction of the inequality changes when we multiply or divide by a negative number. Our task does not require this step; instead, we simply add or subtract the same value across the inequality to maintain the relationship.

In our exercise, we added \( \frac{11}{3} \) to each part of the compound inequality resulting from the absolute value to find the solution. This action preserves the direction of the inequality and results in \( \frac{7}{3} < x < \frac{15}{3} \), which further simplifies down to \( 2\frac{1}{3} < x < 5 \). The important takeaway is the systematic approach to such inequalities—manipulating them step by step while cautiously observing the signs and maintaining the balance across the inequality.

Absolute Value Properties

The absolute value of a number refers to its distance from zero on the number line, disregarding its sign. This is why \( |x| \) is always non-negative. Two key properties to remember are: \( |a| = a \) if \( a \) is non-negative, and \( |a| = -a \) if \( a \) is negative. Additionally, for equations involving absolute values, \( |a| = b \) means \( a \) could be \( b \) or \( -b \) because both would have the same absolute value.

In the context of inequalities like \( |x - a| < b \) we interpret this as \( x \) lying within a distance \( b \) from \( a \) in both the positive and negative directions. This is why it breaks down into the compound inequality \( -b < x - a < b \) during the solving process, implicitly acknowledging that the variable \( x \) could be \( a + b \) or \( a - b \) away from \( a \) on the number line. Understanding these properties makes it easier to interpret and solve absolute value inequalities.