Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ 9-2 b \leq 1 $$

Step-by-Step Solution

Answer

The solution is \( b \geq 4 \), with the smallest integer solution being 4.

Step 1: Isolate the Variable Term

First, we need to isolate the term containing the variable on one side of the inequality. To do this, subtract 9 from both sides of the inequality: \[ 9 - 2b - 9 \leq 1 - 9 \].This simplifies to:\[ -2b \leq -8 \].

Step 2: Solve for the Variable

Divide each side of the inequality by -2, remembering to reverse the inequality sign because we are dividing by a negative number:\[ b \geq 4 \].

Step 3: Interpret the Solution

The inequality \( b \geq 4 \) tells us that \( b \) can be any integer greater than or equal to 4. In this case, the smallest integer solution is 4.

Key Concepts

Integer Solutions

When solving inequalities, it is crucial to understand the concept of integer solutions. An integer is a whole number that can be either positive, negative, or zero but does not include fractions or decimals.

• For instance, numbers like -2, 0, and 3 are integers.
• In our inequality, where we found that \( b \geq 4 \), this implies all possible integer solutions must be 4 or greater.

Additionally, when asked for integer solutions, scrutinize the context to ensure that only whole numbers are considered legitimate solutions.
_In our example, even though a number like 4.5 satisfies the inequality \( b \geq 4 \), it is not considered an integer solution._Understanding integer solutions is vital for correctly interpreting the range of possible answers for inequalities.

Isolation of Variable

In solving inequalities, isolating the variable refers to manipulating the inequality so that the variable of interest is by itself on one side.
This often involves numerous steps, including addition, subtraction, multiplication, or division. Let's break it down:

• First, determine which term contains the variable. In our exercise, the term is \(-2b\).
• To isolate it, subtract any constant from both sides of the inequality to minimize distractions. We subtracted 9 from both sides, leading to \(-2b \leq -8\).

This step makes the inequality simpler and focuses solely on the variable, which further helps in simplifying and solving it. Remember, each action should help you gradually clear the variable from other terms on the same side.

Reversing Inequality Sign

When dealing with inequalities, a critical rule to remember is that the inequality sign must be reversed whenever you multiply or divide by a negative number. This maintains the truth of the inequality. Let's take a closer look:

• In our equation, to solve \(-2b \leq -8\), we divided both sides by \(-2\) to get \(b\) alone.
• This operation requires flipping the inequality sign from \(\leq\) to \(\geq\). Hence, the resulting inequality is \(b \geq 4\).

This might seem counterintuitive initially, but it is essential in maintaining the correct relational state defined by the inequality.
Always be vigilant in keeping track of this necessary step in algebraic manipulations involving negative numbers to ensure your solutions are accurate.