Problem 59 - Algebra and Trigonometry

Question

Using Distances to Solve Absolute Value Inequalities Recall that \(|a-b|\) is the distance between a and \(b\) on the number line. For any number \(x\) , what do \(|x-1|\) and \(|x-3|\) represent? Use this interpretation to solve the inequality \(|x-1|<|x-3|\) geometrically. In general, if \(a

Step-by-Step Solution

Answer

The solution of \(|x-1|<|x-3|\) is the interval \((1, 2)\). Generally, \(|x-a|<|x-b|\) implies \(x\in(a, \frac{a+b}{2})\) for \(a

Step 1: Understanding the Absolute Value

The expression \(|x-1|\) represents the distance of \(x\) from the number 1 on the number line. Similarly, \(|x-3|\) represents the distance of \(x\) from the number 3 on the number line. Therefore, the inequality \(|x-1| < |x-3|\) means the distance of \(x\) from 1 is less than the distance from 3.

Step 2: Interpret Geometrically

To solve \(|x-1| < |x-3|\) geometrically, consider the points 1 and 3 on a number line. The inequality \(|x-1| < |x-3|\) suggests that \(x\) must be closer to 1 than to 3. Visually, this corresponds to the interval that is closer to 1 on the number line. Here, the number line segment between 1 and 3 fits this condition.

Step 3: Solving the Inequality

Geometrically, the region between the points 1 and 3 (which is the interval \((1, 2)\)) fulfills the condition \(|x-1| < |x-3|\). This means \(x\) needs to be nearer to 1 than to 3. Mathematically, this solution is the interval \((1, 2)\).

Step 4: Generalizing the Solution

For the inequality \(|x-a| < |x-b|\), where \(a

Key Concepts

Distance on Number Line

When we talk about distance on a number line, we are referring to the absolute value of the difference between two points. Thus, the expression \(|x-1|\) tells us how far the number \(x\) is from 1, while \(|x-3|\) tells us the distance from 3. The concept of distance on a number line is crucial as it helps us understand how quantities or positions compare with one another visually and numerically.

• Distance provides a way to measure how far apart points are without direction involved.
• In absolute value, negative distances are turned positive, reflecting actual space covered.

Grasping this concept helps in solving many types of problems, especially those involving comparisons of size, such as absolute value inequalities.

Inequality Solution

In the realm of inequalities, solving absolute value inequalities like \(|x-1| < |x-3|\) involves finding which values of \(x\) make the inequality true. The statement \(|x-1| < |x-3|\) means that \(x\) is closer to 1 than it is to 3 on the number line.

• This implies that \(x\) lies between 1 and an average point between 1 and 3, specifically less than midway.
• Mathematically, this solution is captured as the interval \((1, 2)\).

By isolating \(x\) within this interval, the inequality holds true, meaning we've effectively "solved" it by identifying all possible values that satisfy the condition.

Geometric Interpretation

Understanding the geometric interpretation of absolute value inequalities can greatly simplify solving them. When dealing with \(|x-1| < |x-3|\), think of it as describing the location of \(x\) relative to two fixed locations: 1 and 3.

• Geometrically, this inequality suggests that the point for \(x\) is located between the points 1 and 3 on the number line, nearer to 1.
• This understanding is visually represented as the segment between 1 and 2 (exclusive of 2), where \(x\) resides.

This geometric approach turns what might seem like an abstract inequality into a tangible, visual representation on the number line, making it easier to comprehend.

Interval Notation

Interval notation offers a concise way to express a set of numbers that are solutions to an inequality. For the inequality \(|x-1| < |x-3|\), the solution, as determined through geometric reasoning, is the interval \((1, 2)\).

• The use of round brackets "( )" shows that the endpoints 1 and 2 are not included in the solution set.
• This notation is pivotal in communicating mathematical ideas precisely, especially in higher mathematics.

By representing solutions in interval notation, we can convey complex sets of solutions in a straightforward and universally understood format.

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Key Concepts

Distance on Number Line

When we talk about distance on a number line, we are referring to the absolute value of the difference between two points. Thus, the expression \(|x-1|\) tells us how far the number \(x\) is from 1, while \(|x-3|\) tells us the distance from 3. The concept of distance on a number line is crucial as it helps us understand how quantities or positions compare with one another visually and numerically.

• Distance provides a way to measure how far apart points are without direction involved.
• In absolute value, negative distances are turned positive, reflecting actual space covered.

Grasping this concept helps in solving many types of problems, especially those involving comparisons of size, such as absolute value inequalities.

Inequality Solution

In the realm of inequalities, solving absolute value inequalities like \(|x-1| < |x-3|\) involves finding which values of \(x\) make the inequality true. The statement \(|x-1| < |x-3|\) means that \(x\) is closer to 1 than it is to 3 on the number line.

• This implies that \(x\) lies between 1 and an average point between 1 and 3, specifically less than midway.
• Mathematically, this solution is captured as the interval \((1, 2)\).

By isolating \(x\) within this interval, the inequality holds true, meaning we've effectively "solved" it by identifying all possible values that satisfy the condition.

Geometric Interpretation

Understanding the geometric interpretation of absolute value inequalities can greatly simplify solving them. When dealing with \(|x-1| < |x-3|\), think of it as describing the location of \(x\) relative to two fixed locations: 1 and 3.

• Geometrically, this inequality suggests that the point for \(x\) is located between the points 1 and 3 on the number line, nearer to 1.
• This understanding is visually represented as the segment between 1 and 2 (exclusive of 2), where \(x\) resides.

This geometric approach turns what might seem like an abstract inequality into a tangible, visual representation on the number line, making it easier to comprehend.

Interval Notation

Interval notation offers a concise way to express a set of numbers that are solutions to an inequality. For the inequality \(|x-1| < |x-3|\), the solution, as determined through geometric reasoning, is the interval \((1, 2)\).

• The use of round brackets "( )" shows that the endpoints 1 and 2 are not included in the solution set.
• This notation is pivotal in communicating mathematical ideas precisely, especially in higher mathematics.

By representing solutions in interval notation, we can convey complex sets of solutions in a straightforward and universally understood format.