Solve $m-2(m-4) \leq 3m$ For $m$.A. $m \geq -2$B. \$m \leq -2$[/tex\]C. $m \leq 2$D. $m \geq 2$

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Solving Inequalities: A Step-by-Step Guide to Solving the Inequality $m-2(m-4) \leq 3m$

In this article, we will delve into the world of inequalities and learn how to solve them. Specifically, we will focus on solving the inequality $m-2(m-4) \leq 3m$. This type of problem is commonly encountered in algebra and is an essential skill to master for anyone looking to excel in mathematics.

What is an Inequality?

An inequality is a statement that two expressions are not equal. It can be either greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). In this case, we are dealing with the inequality $m-2(m-4) \leq 3m$, which means that the expression on the left-hand side is less than or equal to the expression on the right-hand side.

Step 1: Simplify the Left-Hand Side

To solve the inequality, we need to simplify the left-hand side by combining like terms. We can start by distributing the negative 2 to the terms inside the parentheses:

m−2(m−4)=m−2m+8m-2(m-4) = m-2m+8

Now, we can combine the like terms:

m−2m+8=−m+8m-2m+8 = -m+8

So, the inequality becomes:

−m+8≤3m-m+8 \leq 3m

Step 2: Add m to Both Sides

To isolate the variable m, we need to get all the terms with m on one side of the inequality. We can do this by adding m to both sides:

−m+8+m≤3m+m-m+8+m \leq 3m+m

This simplifies to:

8≤4m8 \leq 4m

Step 3: Divide Both Sides by 4

To solve for m, we need to get rid of the coefficient 4 that is being multiplied by m. We can do this by dividing both sides by 4:

84≤4m4\frac{8}{4} \leq \frac{4m}{4}

This simplifies to:

2≤m2 \leq m

Therefore, the solution to the inequality $m-2(m-4) \leq 3m$ is $m \geq 2$. This means that any value of m that is greater than or equal to 2 will satisfy the inequality.

The correct answer is:

  • A. $m \geq -2$: This is not the correct answer, as the solution to the inequality is $m \geq 2$, not $m \geq -2$.
  • B. $m \leq -2$: This is not the correct answer, as the solution to the inequality is $m \geq 2$, not $m \leq -2$.
  • C. $m \leq 2$: This is not the correct answer, as the solution to the inequality is $m \geq 2$, not $m \leq 2$.
  • D. $m \geq 2$: This is the correct answer.
  • When solving inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that you are simplifying the expressions correctly.
  • When adding or subtracting the same value to both sides of an inequality, the direction of the inequality remains the same.
  • When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality is reversed.
  • Solve the inequality $2x+5 \leq 3x-2$.
  • Solve the inequality $x-3 \geq 2x+1$.
  • Solve the inequality $4x-2 \leq 2x+5$.

In conclusion, solving inequalities requires a step-by-step approach and a solid understanding of algebraic concepts. By following the steps outlined in this article, you can master the art of solving inequalities and become proficient in mathematics. Remember to always simplify the expressions, follow the order of operations, and be mindful of the direction of the inequality. With practice and patience, you will become a pro at solving inequalities in no time!