Problem 17 - Algebra 1: Concepts and Skills

Question

Describe the steps you would use to solve the inequality. $$ 22+3 b \leq-2 $$

Step-by-Step Solution

Answer

The solution of the inequality is \(b \leq -8\).

Step 1: Subtract 22 from Both Sides

In order to isolate the terms involving \(b\), subtract 22 from both sides of the inequality. This results in \(3b \leq -2 - 22\). Simplifying the right side gives \(3b \leq -24\).

Step 2: Divide by 3

Lastly, to solve for \(b\), divide both sides of the inequality by 3. This gives \(b \leq -24 / 3\). Simplification results in \(b \leq -8\).

Key Concepts

Inequality Solutions

Understanding how to solve inequalities is a fundamental skill in algebra, similar to solving equations, but with a couple of additional twists. The goal is to find the values of the variable that make the inequality true.

Inequalities can have multiple solutions since they often indicate a range of values, rather than a single number. For example, the solution to the inequality given in our exercise, which is \(b \leq -8\), tells us that any number less than or equal to -8 will satisfy the condition. Unlike an equation, where we might find a single value for \(b\), the inequality solution here is a continuum of values, extending infinitely in one direction along the number line.

When solving inequalities, it's crucial to remember that if we multiply or divide by a negative number, we must reverse the inequality sign. This specific exercise does not require this step, but it is an important rule to keep in mind for other problems.

Algebraic Steps

Algebra is all about taking logical steps to manipulate expressions in order to solve for an unknown variable. In the context of inequalities, these algebraic steps should be taken carefully to ensure the integrity of the inequality is preserved.

The initial step in our exercise involved subtraction: we subtracted 22 from both sides to eliminate the constant term on the side containing the variable \(b\), making the inequality easier to solve.

Why subtraction matters

Subtraction helps us move closer to isolating the variable, which is essential for finding the solution set for \(b\). After the subtraction, we are left with a cleaner, simpler form of the inequality, allowing us to see more clearly what the next step should be.

Isolation of Variables

Isolating the variable is the key to solving both equations and inequalities. It boils down to having the variable on one side of the inequality and everything else on the other side.

After subtracting 22 from both sides in our inequality example, we had \(3b \leq -24\), which brought us one step closer to isolation of \(b\). However, \(b\) wasn't alone yet—it was still being multiplied by 3.

The Division Step

To fully isolate \(b\), we needed to perform division, which is one of the most critical algebraic operations. Dividing both sides of the inequality by 3 ensured that \(b\) stood by itself on one side, giving us the final solution of \(b \leq -8\). Remember, if we had to divide by a negative number instead, we would have switched the inequality sign to maintain the correct relational order.