Problem 21 - Algebra 1: Concepts and Skills

Question

Check to see if \(b=8\) is or is not a solution of the inequality. $$ 3 b-24>0 $$

Step-by-Step Solution

Answer

No, \(b=8\) is not a solution of the inequality \(3b-24>0\).

Step 1: Plug in the value

Replace \(b\) with 8 in the inequality. It becomes \(3*8 - 24 > 0\). Now simplify it.

Step 2: Simplify

The simplification leads to: \(24 - 24 > 0\).

Step 3: Check if the inequality holds

The simplified form: \(0 > 0\). This inequality does not hold, as zero is not greater than zero.

Key Concepts

Checking Solutions

Checking whether a specific value satisfies an inequality is a straightforward process. It involves substituting the given value into the inequality and determining if the resulting statement is true. This simple exercise can sometimes be trickier than it seems, due to the rearrangements and simplifications involved.
Let's break it down step by step:

• Substitute the Value: Replace the variable in your inequality with the given number.
• Simplify: Perform any operations to simplify the expression on both sides of the inequality.
• Evaluate: Check if the simplified version of the inequality is true.

In our example, substituting 8 for \( b \) in the inequality \(3b - 24 > 0\) results in \(3(8) - 24 > 0\), which simplifies to \(0 > 0\). This indicates that the original inequality is false, meaning \( b = 8 \) is not a solution.

Algebraic Inequalities

Algebraic inequalities are similar to algebraic equations but instead of an equals sign, they use inequality symbols like <, >, ≤, and ≥. These symbols describe a range of possible values that make the inequality true.
Understanding these symbols is crucial:

• "Greater than" (>): Means the left side is larger than the right side.
• "Less than" (<): Indicates the left side is smaller.
• "Greater than or equal to" (≥): The left side is at least as large as the right side.
• "Less than or equal to" (≤): The left side is at most as large.

The aim when working with algebraic inequalities is to find the set of values that satisfy the inequality. You can manipulate inequalities similarly to equations, but be careful with multiplying or dividing by negative numbers as this reverses the inequality symbol.

Inequality Problem Solving

Solving inequality problems requires a blend of algebraic manipulation and logical reasoning. Here are some general steps to follow:

• Identify the Inequality: Determine what type of inequality you're working with.
• Isolate the Variable: Use algebraic techniques to get the variable by itself on one side. Remember, you can add, subtract, multiply, or divide both sides by the same number; however, if you multiply or divide by a negative number, reverse the inequality symbol.
• Check Your Solution: Substitute back into the original inequality to verify it holds true for the found values.

In inequality problem solving, it's important to visualize or draw a number line representation, particularly when dealing with quadratic or compound inequalities. This will help in understanding the range of solutions. Be meticulous in each algebraic step to avoid errors, as even small mistakes can lead to incorrect solutions.