Problem 23 - Algebra 2 and Trigonometry

Question

In \(15-26,\) solve each inequality and write the solution set if the variable is an element of the set of integers. $$ |5-b|+4<9 $$

Step-by-Step Solution

Answer

The solution set is \( \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \).

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression in the inequality. We have \(|5-b| + 4 < 9\). Subtract 4 from both sides to isolate the absolute value: \(|5-b| < 5\).

Step 2: Create Two Inequalities

The inequality \(|5-b| < 5\) means that the expression inside the absolute value is between -5 and 5. This can be split into two inequalities: \(-5 < 5-b < 5\).

Step 3: Solve the Left Inequality

Solve the left inequality \(-5 < 5 - b\) by subtracting 5 from both sides: \(-10 < -b\). Then, multiply all terms by -1, which reverses the inequality sign, giving \(b < 10\).

Step 4: Solve the Right Inequality

Next, solve the right inequality \(5-b < 5\). Subtract 5 from both sides: \(-b < 0\). Multiplying both sides by -1 (and reversing the inequality) gives \(b > 0\).

Step 5: Combine the Solutions

The solutions from the two inequalities \(b < 10\) and \(b > 0\) combine to give \(0 < b < 10\). Since we want integer solutions, this means \(b\) can be 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities are expressions that involve the absolute value of a variable. An absolute value \( |x| \) denotes the distance of \( x \) from zero on the number line, which means it's always non-negative.
In terms of inequalities, this concept helps to understand how far a value can deviate from a central point. For example, the inequality \( |5-b| < 5 \) implies we are looking for values of \( b \) such that the difference between \( b \) and 5 is less than 5 units.
To solve this kind of inequality, you need to split it into two separate inequalities:

• The positive side inequality: \( 5-b < 5 \)
• The negative side inequality: \( 5-b > -5 \)

This gives a range of possible values for \( b \) that satisfy the inequality. Recognizing and setting up these dual inequalities allows you to handle any problem that involves distances or magnitudes within specified limits.

Finding Integer Solutions

Once you've established the range of values for your variable from the split inequalities, as demonstrated in our example by obtaining \( 0 < b < 10 \), the next step focuses on finding integer solutions.
Integer solutions are simply values of the variable that are whole numbers. This often involves identifying all the whole numbers that lie within the range you calculated.
In the given inequality, the valid integer solutions are the numbers between 0 and 10 but not including 0 and 10 themselves. This is because the inequality signs are strictly less than and greater than, rather than inclusive \( \leq \) or \( \geq \).
By listing them out, we find the integer solutions are:

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9

Understanding the demand of integer-only solutions is crucial when moving from theoretical resolution to practical answer seeking, especially when your subject matter involves scenarios where fractional values don't make sense, like counting objects.

Using Inequality Properties

Inequality properties are pivotal in manipulating and solving inequalities effectively. These rules allow us to move terms, simplify expressions, and reveal solutions:

• Addition and subtraction of the same number to/from both sides of an inequality maintain its direction.
• Multiplying or dividing both sides by a positive number keeps the inequality direction unchanged.
• Multiplying or dividing both sides by a negative number requires flipping the inequality sign.

For example, in the step \( -10 < -b \), multiplying through by \(-1\) flips the sign to \( b > 0 \), revealing new information about \( b \).
These properties are not only mathematical tools but also a logical framework for thinking about how constraints and limits behave under various conditions. Using these properties, you can transform and solve inequalities validly, catering to a wide variety of both academic and real-world problem contexts.