Problem 4 - Algebra 1: Concepts and Skills

Question

Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ \frac{3}{4} a<0 $$

Step-by-Step Solution

Answer

The inequality \( \frac{3}{4} a < 0 \) is not a multi-step inequality. It involves a single multiplication operation to solve, yielding the solution \( a < 0 \).

Step 1: Determine the Type of Inequality

Check if the given inequality \( \frac{3}{4} a < 0 \) is a multi-step inequality. It's evident that it is not because it only involves a single multiplication of the variable.

Step 2: Solving the Inequality

Since you are trying to solve for 'a', you should isolate 'a' in the inequality. You can do this by multiplying both sides by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). This gives: \( \frac{3}{4} a * \frac{4}{3} < 0 * \frac{4}{3} \) simplifying to: \( a<0 \).

Key Concepts

Multi-step Inequalities

Multi-step inequalities involve multiple operations to isolate the variable. You might need to add, subtract, multiply, or divide more than once. Identifying whether an inequality is "multi-step" is the starting point of solving it.
In our exercise, \( \frac{3}{4} a < 0 \), we can see it requires only one step, making it a single-step inequality. Therefore, in cases like this, focus on recognizing how many operations are needed to isolate the variable. Here's how you can tell if an inequality is multi-step:

• Are there terms on both sides that need to be moved?
• Do operations involve distribution or combining like terms?
• Do you have more than one arithmetic operation to complete?

If the answer is yes to any of these, it's a multi-step.Understanding the problem type will guide you on how to approach it effectively.

Inequality Solving Steps

Solving inequalities follows a series of logical steps, similar to solving equations, but with one key difference - the direction of the inequality sign can change if you multiply or divide by a negative number.
In any inequality:

• First, simplify both sides as much as possible.
• Use addition or subtraction to eliminate constant terms from one side.
• Apply multiplication or division to remove coefficients from the variable.
• If multiplication or division by a negative number is involved, remember to flip the inequality sign.

For our inequality \( \frac{3}{4} a < 0 \), the step is straightforward as it involves multiplying by only one positive number. These steps guide you methodically through the process, ensuring that you solve the inequality correctly.

Isolation of Variable

The isolation of a variable is about getting the variable alone on one side of the inequality. This is your ultimate goal in solving any inequality or equation. Consider our exercise, \( \frac{3}{4} a < 0 \).Only one step is required: multiply by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). By doing so, you clear the fraction:\[(\frac{3}{4} a) \times (\frac{4}{3}) < 0 \times \frac{4}{3}\]This simplifies directly to \( a < 0 \).It's essential to:

• Perform the same operation across the entire inequality to maintain balance.
• Remember any special rules for inequalities, like reversing the inequality sign when multiplying or dividing by negatives.

Achieving the isolation efficiently requires a clear understanding of these principles. Mastery of variables isolation will simplify many algebra problems.