Problem 42 - A Graphical Approach to College Algebra

Question

Solve each group of equations and inequalities analytically. (a) \(|-9-3 x|=6\) (b) \(|-9-3 x| \geq 6\) (c) \(|-9-3 x| \leq 6\)

Step-by-Step Solution

Answer

(a) \( x = -5 \) or \( x = -1 \); (b) \( x \leq -5 \) or \( x \geq -1 \); (c) \( -5 \leq x \leq -1 \).

Step 1: Understanding Absolute Value Equation (a)

We need to solve the equation \( |-9 - 3x| = 6 \). This splits into two separate equations: \( -9 - 3x = 6 \) and \( -9 - 3x = -6 \) due to the property of absolute values.

Step 2: Solving First Equation of (a)

First, solve \( -9 - 3x = 6 \). Adding 9 to both sides, we get \( -3x = 15 \). Dividing both sides by -3 gives \( x = -5 \).

Step 3: Solving Second Equation of (a)

Next, solve \( -9 - 3x = -6 \). Add 9 to both sides to get \( -3x = 3 \). Dividing by -3, we obtain \( x = -1 \).

Step 4: Understanding Absolute Value Inequality (b)

For \( |-9 - 3x| \geq 6 \), we break it into two conditions based on the definition of absolute value: \( -9 - 3x \geq 6 \) or \( -9 - 3x \leq -6 \).

Step 5: Solving First Condition of (b)

Solve \( -9 - 3x \geq 6 \). Add 9 to both sides to obtain \( -3x \geq 15 \). Divide by -3, remembering to reverse the inequality, to get \( x \leq -5 \).

Step 6: Solving Second Condition of (b)

Solve \( -9 - 3x \leq -6 \). Add 9 to both sides to get \( -3x \leq 3 \). Divide by -3 and reverse the inequality to find \( x \geq -1 \).

Step 7: Combine Solutions for (b)

The solution for \( |-9 - 3x| \geq 6 \) is \( x \leq -5 \) or \( x \geq -1 \).

Step 8: Understanding Absolute Value Inequality (c)

For \( |-9 - 3x| \leq 6 \), we use the definition of absolute value to set up the compound inequality: \(-6 \leq -9 - 3x \leq 6\).

Step 9: Solving the Compound Inequality for (c)

Start with \(-6 \leq -9 - 3x \). Add 9 to obtain \( 3 \leq -3x \). Divide by -3, reversing the inequality, to get \( x \leq -1 \).Next, solve \( -9 - 3x \leq 6 \). Add 9 to both sides to get \( -3x \leq 15 \). Divide by -3 to obtain \( x \geq -5 \).

Step 10: Combine Solutions for (c)

The solution for the compound inequality \( |-9 - 3x| \leq 6 \) is \( -5 \leq x \leq -1 \).

Key Concepts

Absolute Value Equations

Absolute value equations can be a bit tricky at first, but they're all about distance from zero on the number line. The absolute value of a number is always positive, because it represents how far that number is from zero, no matter which direction. So, when you encounter an equation like \(|-9 - 3x| = 6\), you're dealing with values that can be both positive and negative inside the absolute value brackets.

To solve an absolute value equation, you need to split it into two separate equations. For example, \(|-9 - 3x| = 6\) becomes two equations:

• \(-9 - 3x = 6\)
• \(-9 - 3x = -6\)

Each of these equations is solved independently. Solving \(-9 - 3x = 6\) entails adding 9 to both sides, resulting in \(-3x = 15\). Then, divide by -3, giving \(x = -5\). Similarly, solving \(-9 - 3x = -6\) gives you \(x = -1\).

This technique ensures you account for both scenarios where the expression inside the absolute value could be positive or negative.

Inequalities

Inequalities describe a range of values rather than a single solution. Solving inequalities involves finding all the values of a variable that make the inequality true. When dealing with absolute value inequalities, they split into two conditions too! Let's take a look at \(|-9 - 3x| \geq 6\).

Just like absolute value equations, absolute value inequalities can turn into two expressions:

• \(-9 - 3x \geq 6\)
• \(-9 - 3x \leq -6\)

For \(-9 - 3x \geq 6\), add 9 to both sides, leading to \(-3x \geq 15\). Dividing by -3 causes the inequality to flip, giving \(x \leq -5\). For \(-9 - 3x \leq -6\), adding 9 yields \(-3x \leq 3\), and dividing by -3 results in \(x \geq -1\).

Combine these to find \(x \leq -5\) or \(x \geq -1\). This solution means any x value less than or equal to -5, or greater than or equal to -1, satisfies the inequality.

Compound Inequalities

Compound inequalities involve more than one inequality dealing with the same variable. Specifically, a sandwich-like structure shows up, like in \(|-9 - 3x| \leq 6\). This results in a single compound inequality composed of two inequalities.

The expression \(|-9 - 3x| \leq 6\) can be expressed as:

• \(-6 \leq -9 - 3x \leq 6\)

This states that the expression \(-9 - 3x\) must lie between -6 and 6.

To solve, first deal with the left side by adding 9, giving \(3 \leq -3x\). Divide by -3, and flip the sign to obtain \(x \leq -1\). Next, solve the right side by adding 9, yielding \(-3x \leq 15\). Again, divide by -3, resulting in \(x \geq -5\).

The solution for \(|-9 - 3x| \leq 6\) is the intersection of these solutions, or \(-5 \leq x \leq -1\). This compound inequality shows all the x values that work for both parts of the original inequality.