Solving Absolute Value Equations A Step-by-Step Guide For -2|m+1|+9=7
Hey guys! 👋 Today, we're diving deep into the fascinating world of absolute value equations and tackling a specific problem: -2|m+1|+9=7. Absolute value equations might seem intimidating at first, but don't worry, we'll break it down step-by-step, making it super easy to understand. We'll not only solve the equation but also check our answer to ensure accuracy. So, buckle up, grab your thinking caps, and let's get started!
Understanding Absolute Value
Before we jump into solving the equation, it's crucial to have a solid grasp of what absolute value actually means. In simple terms, the absolute value of a number is its distance from zero on the number line. Distance is always non-negative, so the absolute value of a number will always be either zero or positive. We represent the absolute value using vertical bars: | |. For instance, |3| = 3 because 3 is 3 units away from zero, and |-3| = 3 because -3 is also 3 units away from zero. Think of it like this: absolute value strips away the sign (positive or negative) and gives you the magnitude of the number.
Key takeaways about absolute value:
- The absolute value of a positive number is the number itself.
- The absolute value of a negative number is the positive version of that number.
- The absolute value of zero is zero.
Steps to Solve Absolute Value Equations
Alright, now that we're clear on what absolute value is, let's outline the general steps for solving absolute value equations. This will give us a roadmap for tackling our specific problem.
- Isolate the Absolute Value Expression: This is the most important first step. We need to get the absolute value expression (the part inside the vertical bars) all by itself on one side of the equation. This usually involves using inverse operations (addition, subtraction, multiplication, division) to move other terms away from the absolute value.
- Set Up Two Equations: This is where the magic happens! Because absolute value deals with distance from zero, there are usually two possibilities to consider. If |x| = a (where a is a non-negative number), then either x = a or x = -a. We'll set up two separate equations representing these two possibilities.
- Solve Each Equation: Now we have two simple equations to solve. We'll use standard algebraic techniques to isolate the variable in each equation.
- Check Your Solutions: This is a crucial step often overlooked! We need to plug each solution back into the original absolute value equation to make sure it works. Absolute value equations can sometimes produce extraneous solutions (solutions that don't actually satisfy the original equation), so checking is a must.
Solving -2|m+1|+9=7: A Step-by-Step Walkthrough
Okay, with our roadmap in place, let's dive into solving our equation: -2|m+1|+9=7.
Step 1: Isolate the Absolute Value Expression
Remember, our first goal is to get the absolute value part, |m+1|, all by itself. To do this, we'll start by subtracting 9 from both sides of the equation:
-2|m+1|+9-9 = 7-9
This simplifies to:
-2|m+1| = -2
Now, we need to get rid of the -2 that's multiplying the absolute value. We'll divide both sides by -2:
(-2|m+1|)/-2 = -2/-2
This gives us:
|m+1| = 1
Great! We've successfully isolated the absolute value expression.
Step 2: Set Up Two Equations
Now comes the fun part! Since |m+1| = 1, this means that the expression inside the absolute value, (m+1), is either 1 unit away from zero in the positive direction or 1 unit away from zero in the negative direction. This gives us two possibilities:
- m + 1 = 1
- m + 1 = -1
We've now transformed our single absolute value equation into two separate linear equations.
Step 3: Solve Each Equation
Let's solve each equation individually.
Equation 1: m + 1 = 1
To solve for m, we subtract 1 from both sides:
m + 1 - 1 = 1 - 1
This gives us:
m = 0
So, our first potential solution is m = 0.
Equation 2: m + 1 = -1
Again, we subtract 1 from both sides to isolate m:
m + 1 - 1 = -1 - 1
This simplifies to:
m = -2
Our second potential solution is m = -2.
Step 4: Check Your Solutions
Alright, we've got two potential solutions: m = 0 and m = -2. Now, we absolutely must check them in the original equation, -2|m+1|+9=7, to make sure they work.
Checking m = 0:
Substitute m = 0 into the original equation:
-2|0+1|+9=7
Simplify:
-2|1|+9=7
-2(1)+9=7
-2+9=7
7=7
This is true! So, m = 0 is a valid solution.
Checking m = -2:
Substitute m = -2 into the original equation:
-2|-2+1|+9=7
Simplify:
-2|-1|+9=7
-2(1)+9=7
-2+9=7
7=7
This is also true! So, m = -2 is also a valid solution.
The Solutions
We've done it! We've successfully solved the absolute value equation -2|m+1|+9=7. Our solutions are:
- m = 0
- m = -2
Why Checking Solutions is Crucial
I can't stress enough how important it is to always check your solutions when dealing with absolute value equations. Let's take a moment to understand why. Absolute value equations often involve splitting the problem into two cases, and sometimes, one or both of those cases can lead to a solution that doesn't actually work in the original equation. These are called extraneous solutions.
Extraneous solutions arise because the process of squaring both sides of an equation (which is sometimes used to eliminate the absolute value) can introduce solutions that satisfy the squared equation but not the original equation. By checking our solutions, we weed out any extraneous ones and ensure we only keep the valid solutions.
Real-World Applications of Absolute Value
You might be wondering,
