Problem 10 - Algebra 1: Concepts and Skills

Question

Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ 3 m+2 \leq 7 m $$

Step-by-Step Solution

Answer

No, the given inequality is not a multi-step inequality. The solution to the inequality is \(m \geq 0.5\).

Step 1: Identify the Type of Inequality

The given inequality is \(3m + 2 \leq 7m\). It can be noticed that the inequality only requires the subtraction operation to solve, so it is not a multi-step inequality.

Step 2: Rearrange the Inequality

To solve the inequality, the variable 'm' should be isolated on one side of the inequality. For this, subtract \(3m\) from both sides of the inequality to obtain: \(2 \leq 4m\).

Step 3: Solve the Inequality

Now, to get the isolated 'm', divide both sides of the equation by 4, which yields: \(m \geq 0.5\).

Key Concepts

Multi-Step Inequality

A multi-step inequality is one that requires several operations to isolate the variable and find the solution. These operations may include addition, subtraction, multiplication, or division, and may involve combining like terms or using the distributive property.
In multi-step inequalities, each operation must be performed following a specific order of operations to ensure that the inequality is correctly maintained. This is somewhat similar to solving multi-step equations, where every step needs to be carefully executed to maintain the balance of both sides of the inequality.
Identifying a multi-step inequality involves recognizing whether more than one arithmetic operation is needed for simplification. In the exercise given, the inequality \(3m + 2 \leq 7m\) doesn't require multiple arithmetic operations, which means it is not a multi-step inequality. Only a couple of straightforward steps are necessary to solve it, as it doesn't involve any additional terms or the need for factoring.

Inequality Solution Steps

Solving inequalities involves a methodical approach to ensuring the variable is isolated correctly. Here’s a brief overview of the solution steps involved, specifically for the given inequality.

• **Identify the type of inequality:** Determine whether there are multiple operations needed. The inequality \(3m + 2 \leq 7m\) only requires subtraction and division.
• **Rearrange the inequality:** Isolate the variable on one side by subtracting or adding terms as necessary. Subtracting \(3m\) from both sides gives us \(2 \leq 4m\).
• **Solve the inequality:** Once you have a simpler inequality, divide or multiply both sides to solve for the variable. In this example, dividing both sides by 4 isolates 'm', resulting in \(m \geq 0.5\).

It’s important to maintain the inequality sign appropriately through each step. Additionally, remember that when multiplying or dividing by a negative number, the inequality sign reverses.

Subtracting Variables in Inequalities

Subtracting variables in inequalities is a common step, particularly when you need to isolate the variable on one side of the inequality. This process follows similar rules to subtracting numbers while ensuring that the inequality is preserved.
For the inequality \(3m + 2 \leq 7m\), you want all the 'm' terms on one side. By subtracting \(3m\) from both sides, the inequality becomes \(2 \leq 4m\). This step is crucial for shifting all terms involving 'm' to one side, allowing you to isolate 'm' in subsequent steps.
When performing subtraction:

• Identify similar terms to subtract.
• Subtract the coefficients of identical variables across the inequality.
• Keep the inequality sign facing the same way unless the subtraction involves terms that would imply reversing the inequality (not applicable here).

Remember, rearranging through subtraction is vital for keeping inequalities manageable and leading to a straightforward solution. Careful execution of this step ensures the accurate solving of inequalities.