Introduction
Welcome to today's blog article where we will be exploring the topic of solving multi-step inequalities. Inequalities are mathematical statements that compare two values and indicate whether one is greater than or less than the other. Multi-step inequalities involve multiple operations and require careful analysis to find the solution. Understanding and being able to solve these inequalities is an essential skill in algebra and can be applied to various real-life situations. In this article, we will break down the process of solving multi-step inequalities into five steps and provide examples to help you grasp the concept more effectively.
Step 1: Simplify the Inequality
The first step in solving a multi-step inequality is to simplify it as much as possible. This involves combining like terms, distributing, and performing any necessary operations to reduce the complexity of the inequality. By simplifying the inequality, we can make the solving process more manageable and eliminate any unnecessary steps. Let's consider an example:
Example:
2(x + 3) - 5 < 3x + 7
To simplify this inequality, we must distribute the 2 to both terms inside the parentheses:
2x + 6 - 5 < 3x + 7
Next, we combine like terms:
2x + 1 < 3x + 7
Step 2: Move Variables to One Side
In the second step, we want to isolate the variable on one side of the inequality. To do this, we need to move all terms with the variable to one side and the constant terms to the other side. Let's continue with our example:
Example:
2x + 1 < 3x + 7
To move the 3x term to the left side, we subtract 3x from both sides:
2x - 3x + 1 < 3x - 3x + 7
Simplifying further:
-x + 1 < 7
Step 3: Combine Like Terms
After moving the variables to one side, we need to combine like terms to simplify the inequality further. This step helps us consolidate the terms and make it easier to analyze the inequality. Let's proceed with our example:
Example:
-x + 1 < 7
Combining like terms:
-x + 1 - 1 < 7 - 1
Simplifying:
-x < 6
Step 4: Divide or Multiply by a Negative Number
If the coefficient of the variable is negative, we need to divide or multiply both sides of the inequality by a negative number to maintain the inequality's direction. This step ensures that the solution remains accurate. Let's apply this step to our ongoing example:
Example:
-x < 6
In this case, the coefficient of x is -1, so we divide both sides by -1:
-x / -1 > 6 / -1
Remember that when dividing or multiplying by a negative number, the inequality sign flips:
x > -6
Step 5: Finalize the Solution
In the final step, we interpret the solution and write it in an appropriate format. For inequalities, we represent the solution using interval notation or as a number line graph. Let's conclude our example:
Example:
x > -6
The solution to this inequality is all values of x that are greater than -6. We can represent this solution on a number line:
[ -6, ∞ )
Conclusion
Solving multi-step inequalities is a crucial skill in algebra that allows us to compare and analyze mathematical statements. By following the five steps outlined in this article, you can simplify and solve multi-step inequalities effectively. Remember to simplify the inequality, move variables to one side, combine like terms, divide or multiply by a negative number if necessary, and finalize the solution using interval notation or a number line graph. Practice applying these steps to various examples to enhance your understanding and proficiency in solving multi-step inequalities. With time and practice, you'll become more confident and adept at tackling these types of problems.