Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use the geometric interpretation of absolute value to solve the following equations and inequalities. Display the solutions on a number line. (a) \(|x+3|<2\) (b) \(|x-5| \leq 3\) (c) \(|x-a|=b, \quad\) where \(a\) and \(b\) are positive (d) \(|x+a| \leq a, \quad\) where \(a\) is positive

Step-by-Step Solution

Answer

The solutions for the equations/inequalities are: (a) -5

Step 1: Solve the equation \(|x+3|<2\)

To solve \(|x+3|<2\), split the inequality into two parts: \(x+3<2\) and \(-(x+3)<2\).Solving these gives \(x<-1\) and \(x>-5\). Thus, the solution for \(x\) in this inequality is \(-5

Step 2: Solve the inequality \(|x-5|\leq 3\)

Split the inequality \(|x-5|\leq 3\) into two parts: \(x-5\leq 3\) and \(-(x-5)\leq 3\).Solving these gives \(x\leq 8\) and \(x\geq 2\).This means the solution for \(x\) in this inequality is \(2\leq x\leq 8\). On a number line, this would be represented as a closed interval from 2 to 8.

Step 3: Solve the equation \(|x-a|=b\)

The equation \(|x-a|=b\) splits into two equations: \(x-a=b\) and \(x-a=-b\).Solving these results in \(x=a+b\), and \(x=a-b\). This means that 'x' can either be \(a+b\) or \(a-b\). On a number line, 'x' will be represented as two distinct points, \(a-b\) and \(a+b\).

Step 4: Solve the inequality \(|x+a|\leq a\)

The inequality \(|x+a|\leq a\) splits into two inequalities: \(x+a\leq a\) and \(-(x+a)\leq a\).Solving these gives \(x\leq 0\) and \(x\geq -2a\). This means \(x\) in this inequality can range from \(-2a\) to 0, i.e., \(-2a\leq x\leq 0\).On a number line, this would be represented as a closed interval from \(-2a\) to 0.

Key Concepts

Absolute Value

Absolute value simply measures how far a number is from zero on the number line. It doesn't care about direction — only distance. Picture absolute value as the length of a line starting at zero and ending at the number itself. For example, both 3 and -3 have an absolute value of 3 because they are both 3 units away from zero but in opposite directions. When dealing with equations or inequalities, the absolute value affects how you approach finding solutions.

• For equations like \(|x-a|=b\), solve as two separate equations: \(x-a=b\) and \(x-a=-b\).
• For inequalities like \(|x+a|\leq a\), split it into \(x+a\leq a\) and \(-(x+a)\leq a\).

Understanding absolute value helps in measuring distances and applying it to solve real-world problems.

Inequalities

Inequalities express a range of values instead of a single answer. They are conditions showing how one side compares to another, represented by symbols like \(<\), \( \leq \), or \( \geq \).
For instance, \(|x+3|<2\) implies two scenarios: one where \(x+3\) is less than 2 and another where \(x+3\) is more than -2. Solving these gives you a range of possible values for \(x\).
Each inequality is treated separately, ensuring that every potential solution is identified. You always solve twice — once assuming the positive side and once for the negative side.

• An open interval, like in \(-5 < x < -1\), means endpoints are not included.
• A closed interval, like \(2 \leq x \leq 8\), includes both endpoints.

Mastering inequalities allows you to solve for variable ranges accurately.

Number Line

A number line is a simple visual tool that helps represent numbers and their relationships. It's like a ruler where each point corresponds to a number. When solving absolute value equations or inequalities, the number line makes it easier to visualize and understand the solutions.
Suppose we need to represent \(-5 < x < -1\) from \(|x+3|<2\). On the number line, you'll draw an open interval between -5 and -1, symbolizing that these numbers are close but don’t include -5 and -1 themselves.

• Open circles depict values not included.
• Closed circles signify values that are part of the solution set.

Using the number line enhances comprehension by showing where solutions fall relative to other numbers.

Geometric Interpretation

Geometric interpretation involves viewing mathematical concepts through a visual lens. When you think of absolute values geometrically, you imagine distances on the number line.
For example, the equation \(|x-a|=b\) translates to finding two points, \(x=a+b\) and \(x=a-b\), each a distance of 'b' from 'a'. These solutions become two dots on the number line equidistant from a central point.
For absolute value inequalities like \(|x-5| \leq 3\), geometric interpretation tells us we're looking for all values of 'x' that are a distance of 3 or less from 5, effectively forming a closed interval.

• This helps visualize solutions as intervals and points.
• It also shows the symmetry inherent in absolute values around the center point 'a'.

Geometric interpretation simplifies understanding by converting abstract ideas into tangible visual stories.

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Problem 4

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Problem 5

Other exercises in this chapter

Problem 4

Consider the statement "Colleges report that their tuition increases are slowing down." Suppose we set \(t=0\) to be three years before this statement was made

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Problem 4

For Problems 1 through 7, give exact answers, not numerical approximations. A rectangle is 3 meters long and 2 meters high. How long is the diagonal?

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Problem 5

The Boston Globe (August 31,1999, p. 1 ) reports: "While the number of AIDS deaths continues to drop nationally, the rapid rate of decline that had been attribu

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Problem 5

For Problems 1 through 7, give exact answers, not numerical approximations. Solve: \(x^{2}+1=6\)

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Key Concepts

Absolute Value

Absolute value simply measures how far a number is from zero on the number line. It doesn't care about direction — only distance. Picture absolute value as the length of a line starting at zero and ending at the number itself. For example, both 3 and -3 have an absolute value of 3 because they are both 3 units away from zero but in opposite directions. When dealing with equations or inequalities, the absolute value affects how you approach finding solutions.

• For equations like \(|x-a|=b\), solve as two separate equations: \(x-a=b\) and \(x-a=-b\).
• For inequalities like \(|x+a|\leq a\), split it into \(x+a\leq a\) and \(-(x+a)\leq a\).

Understanding absolute value helps in measuring distances and applying it to solve real-world problems.

Inequalities

Inequalities express a range of values instead of a single answer. They are conditions showing how one side compares to another, represented by symbols like \(<\), \( \leq \), or \( \geq \).
For instance, \(|x+3|<2\) implies two scenarios: one where \(x+3\) is less than 2 and another where \(x+3\) is more than -2. Solving these gives you a range of possible values for \(x\).
Each inequality is treated separately, ensuring that every potential solution is identified. You always solve twice — once assuming the positive side and once for the negative side.

• An open interval, like in \(-5 < x < -1\), means endpoints are not included.
• A closed interval, like \(2 \leq x \leq 8\), includes both endpoints.

Mastering inequalities allows you to solve for variable ranges accurately.

Number Line

A number line is a simple visual tool that helps represent numbers and their relationships. It's like a ruler where each point corresponds to a number. When solving absolute value equations or inequalities, the number line makes it easier to visualize and understand the solutions.
Suppose we need to represent \(-5 < x < -1\) from \(|x+3|<2\). On the number line, you'll draw an open interval between -5 and -1, symbolizing that these numbers are close but don’t include -5 and -1 themselves.

• Open circles depict values not included.
• Closed circles signify values that are part of the solution set.

Using the number line enhances comprehension by showing where solutions fall relative to other numbers.

Geometric Interpretation

Geometric interpretation involves viewing mathematical concepts through a visual lens. When you think of absolute values geometrically, you imagine distances on the number line.
For example, the equation \(|x-a|=b\) translates to finding two points, \(x=a+b\) and \(x=a-b\), each a distance of 'b' from 'a'. These solutions become two dots on the number line equidistant from a central point.
For absolute value inequalities like \(|x-5| \leq 3\), geometric interpretation tells us we're looking for all values of 'x' that are a distance of 3 or less from 5, effectively forming a closed interval.

• This helps visualize solutions as intervals and points.
• It also shows the symmetry inherent in absolute values around the center point 'a'.

Geometric interpretation simplifies understanding by converting abstract ideas into tangible visual stories.