Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Solve each group of equations and inequalities analytically. (a) \(|x+4|=9\) (b) \(|x+4|>9\) (c) \(|x+4|<9\)

Step-by-Step Solution

Answer

(a) \(x=5\) or \(x=-13\); (b) \(x>5\) or \(x<-13\); (c) \(-13

Step 1: Understand the Absolute Value Equation (a)

We start with the equation \(|x+4|=9\). This equation implies two possible scenarios: either \(x+4=9\) or \(x+4=-9\) because the absolute value of a number is equal to 9 if the number is either 9 or -9.

Step 2: Solve the Absolute Value Equation (a)

For the first scenario \(x+4=9\), solve it by subtracting 4 from both sides to get \(x=5\). In the second scenario \(x+4=-9\), solve it similarly by subtracting 4 from both sides to get \(x=-13\). Hence, the solutions are \(x=5\) and \(x=-13\).

Step 3: Understand the Absolute Value Inequality (b)

Next, consider \(|x+4|>9\). This inequality means that the expression inside the absolute value, \(x+4\), is either greater than 9 or less than -9.

Step 4: Solve the Absolute Value Inequality (b)

First, solve \(x+4>9\) by subtracting 4 from both sides to get \(x>5\). Then, solve \(x+4<-9\) by subtracting 4 from both sides to get \(x<-13\). Hence, the solution is \(x>5\) or \(x<-13\).

Step 5: Understand the Absolute Value Inequality (c)

Finally, consider \(|x+4|<9\). This inequality implies that \(x+4\) should be between -9 and 9 exclusive.

Step 6: Solve the Absolute Value Inequality (c)

We write two inequalities: \(x+4<9\) and \(x+4>-9\). Solving \(x+4<9\) gives \(x<5\), and solving \(x+4>-9\) gives \(x>-13\). Hence, the solution is \(-13

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value signs being greater than or less than a certain number. These are slightly different from regular equations because they involve ranges of values rather than specific solutions.
In simple terms, the absolute value of \(|x + 4| > 9\) means that the distance of \(x + 4\) from 0 is greater than 9. For inequalities like these, you consider two cases:

• The expression inside is greater than 9.
• Or the expression is less than -9.

This is because an absolute value is always a non-negative number, so the expression could be on either side of zero. The same principle applies if your inequality is \(|x + 4| < 9\), except you're looking for a range of numbers between -9 and 9 on a number line. Understanding these scenarios helps in solving such inequalities effectively.

Step-by-Step Guide to Solving Equations with Absolute Values

When solving absolute value equations, the goal is to isolate the absolute value expression and set up equivalent linear equations, since they represent different possibilities.
For example, consider \(|x+4| = 9\).
You need to break it down into two simpler equations:

• \(x + 4 = 9\)
• \(x + 4 = -9\)

Solving these gives you specific numeric solutions—subtract 4 from both sides to find that \(x = 5\) or \(x = -13\).
This is because both 9 and -9 have an absolute value of 9. Always remember that an absolute value equation like this splits into two possibilities based on the definition of absolute values.

Solutions to Inequality Problems

Absolute value inequalities provide solutions that express a range of numbers rather than pinpoint values. For instance, solving \(|x + 4| > 9\) splits into:

• \(x + 4 > 9\)
• \(x + 4 < -9\)

For the first part, solving gives \(x > 5\). For the second, solving gives \(x < -13\).
As a result, the solution includes two different regions: one greater than 5 and one less than -13. This indicates all possible values that satisfy the inequality.
So the solution is \(x > 5\) or \(x < -13\). The same approach applies for \(|x + 4| < 9\), which you solve by determining that \(xe10\) falls between -13 and 5.

Analytical Methods for Solving Related Problems

Solving absolute value equations and inequalities analytically involves understanding and manipulating expressions by removing the absolute value aspect through case analysis.

• For an equation like \(|x + 4| = 9\), use logic to either boil it down to a simpler problem of \(x + 4 = 9\) or \(x + 4 = -9\).
• With inequalities, such as \(|x + 4| > 9\), split it into \(x + 4 > 9\) or \(x + 4 < -9\) to determine the suitable ranges for \(x\).

These systems demand you deeply understand the absolute value properties—essentially, the 'distance' component in math.
Approaching these problems analytically helps equip students to tackle issues that rely on logic and a systematic breakdown of expressions into manageable parts. Always double-check your solutions by plugging them back into the original conditions to see if they logically fulfill all aspects of the problem you started with.

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Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value signs being greater than or less than a certain number. These are slightly different from regular equations because they involve ranges of values rather than specific solutions.
In simple terms, the absolute value of \(|x + 4| > 9\) means that the distance of \(x + 4\) from 0 is greater than 9. For inequalities like these, you consider two cases:

• The expression inside is greater than 9.
• Or the expression is less than -9.

This is because an absolute value is always a non-negative number, so the expression could be on either side of zero. The same principle applies if your inequality is \(|x + 4| < 9\), except you're looking for a range of numbers between -9 and 9 on a number line. Understanding these scenarios helps in solving such inequalities effectively.

Step-by-Step Guide to Solving Equations with Absolute Values

When solving absolute value equations, the goal is to isolate the absolute value expression and set up equivalent linear equations, since they represent different possibilities.
For example, consider \(|x+4| = 9\).
You need to break it down into two simpler equations:

• \(x + 4 = 9\)
• \(x + 4 = -9\)

Solving these gives you specific numeric solutions—subtract 4 from both sides to find that \(x = 5\) or \(x = -13\).
This is because both 9 and -9 have an absolute value of 9. Always remember that an absolute value equation like this splits into two possibilities based on the definition of absolute values.

Solutions to Inequality Problems

Absolute value inequalities provide solutions that express a range of numbers rather than pinpoint values. For instance, solving \(|x + 4| > 9\) splits into:

• \(x + 4 > 9\)
• \(x + 4 < -9\)

For the first part, solving gives \(x > 5\). For the second, solving gives \(x < -13\).
As a result, the solution includes two different regions: one greater than 5 and one less than -13. This indicates all possible values that satisfy the inequality.
So the solution is \(x > 5\) or \(x < -13\). The same approach applies for \(|x + 4| < 9\), which you solve by determining that \(xe10\) falls between -13 and 5.

Analytical Methods for Solving Related Problems

Solving absolute value equations and inequalities analytically involves understanding and manipulating expressions by removing the absolute value aspect through case analysis.

• For an equation like \(|x + 4| = 9\), use logic to either boil it down to a simpler problem of \(x + 4 = 9\) or \(x + 4 = -9\).
• With inequalities, such as \(|x + 4| > 9\), split it into \(x + 4 > 9\) or \(x + 4 < -9\) to determine the suitable ranges for \(x\).

These systems demand you deeply understand the absolute value properties—essentially, the 'distance' component in math.
Approaching these problems analytically helps equip students to tackle issues that rely on logic and a systematic breakdown of expressions into manageable parts. Always double-check your solutions by plugging them back into the original conditions to see if they logically fulfill all aspects of the problem you started with.