Solve $3|x+1|+3<7$ Find A And B

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Hey guys! Today, we're diving into solving an inequality involving absolute values. Inequalities might seem tricky at first, but don't worry, we'll break it down step by step. Our specific problem is: $3|x+1|+3<7$. The answer will be in the form $A < x < B$, and our mission is to find the values of $A$ and $B$. So, let's grab our mathematical toolkit and get started!

Understanding Absolute Value Inequalities

Before we jump into the nitty-gritty, it's super important to understand what absolute value actually means. The absolute value of a number is its distance from zero. Think of it like this: $|5|$ is 5 because 5 is five units away from zero. And $|-5|$ is also 5, because -5 is also five units away from zero. This “distance” concept is key when we're dealing with inequalities.

Now, when we have an inequality with an absolute value, like our $3|x+1|+3<7$, it means we're looking for all the values of x that make the expression inside the absolute value fall within a certain range. To solve these, we're essentially going to split the problem into two separate cases. Why two cases? Because the expression inside the absolute value can be either positive or negative, and we need to consider both possibilities.

The secret to conquering absolute value inequalities lies in remembering this split. We transform the single inequality into a compound inequality. This means we'll have two inequalities that we need to solve, and the solutions to both will give us the final answer. One inequality will deal with the scenario where the expression inside the absolute value is positive or zero, and the other will deal with the scenario where it's negative. This might sound a bit confusing right now, but trust me, it'll become crystal clear as we work through our example!

Also, remember that whatever you do to one side of the inequality, you've gotta do to the other. This keeps the whole equation balanced and ensures we're on the right track to finding the correct solution. Just like with regular equations, we can add, subtract, multiply, and divide – but with one crucial caveat: multiplying or dividing by a negative number flips the direction of the inequality sign. Keep that in the back of your mind as we proceed!

So, buckle up! We've got the conceptual groundwork laid, and now we're ready to roll up our sleeves and tackle the actual problem. Remember, it's all about breaking it down into manageable steps and keeping that absolute value definition in mind. Let’s do this!

Step 1: Isolate the Absolute Value

Our first order of business is to isolate the absolute value term. This means we want to get $|x+1|$ all by itself on one side of the inequality. Currently, we have $3|x+1|+3<7$. To get rid of that pesky '+3', we'll subtract 3 from both sides of the inequality. Remember, what we do to one side, we must do to the other to maintain balance! So:

$3|x+1|+3 - 3 < 7 - 3$

This simplifies beautifully to:

$3|x+1| < 4$

Awesome! We're making progress. But we're not quite there yet. We still have that '3' multiplying the absolute value. To get rid of it, we'll divide both sides of the inequality by 3. Since 3 is a positive number, we don't need to worry about flipping the inequality sign (phew!). This gives us:

rac{3|x+1|}{3} < rac{4}{3}

Which simplifies to:

|x+1| < rac{4}{3}

Excellent! We've successfully isolated the absolute value. This is a crucial step because now we can clearly see the condition we need to work with. The inequality |x+1| < rac{4}{3} is telling us that the distance between $x+1$ and zero must be less than rac{4}{3}. This is the key to unlocking the next stage of the solution.

Isolating the absolute value is like clearing the fog – it lets us see the core of the problem. Now that we have |x+1| < rac{4}{3}, we're perfectly positioned to split this into our two cases and solve for x. Remember, this step is all about simplifying the original inequality into a form we can work with. So, pat yourself on the back – you've nailed the first major hurdle!

Step 2: Split into Two Cases

This is where the magic happens! Now that we have our isolated absolute value inequality, |x+1| < rac{4}{3}, it's time to split it into those two cases we talked about earlier. Remember, absolute value means considering both the positive and negative possibilities of the expression inside the absolute value bars.

Case 1: The expression inside the absolute value is positive or zero.

In this case, we can simply remove the absolute value bars. So, |x+1| < rac{4}{3} becomes:

x+1 < rac{4}{3}

This is a straightforward inequality that we can solve for x. We'll tackle that in the next step.

Case 2: The expression inside the absolute value is negative.

When the expression inside the absolute value is negative, we need to be a bit careful. Remember, the absolute value turns a negative into a positive. So, to account for this, we'll take the negative of the expression inside the absolute value and set it less than rac{4}{3}. This means we'll have:

-(x+1) < rac{4}{3}

This is equivalent to saying that the opposite of $x+1$ is less than rac{4}{3}. This might seem a bit tricky, but it's crucial for capturing all the possible solutions.

Another way to think about this case is to multiply both sides of the inequality |x+1| < rac{4}{3} by -1 and also flip the direction of the inequality. This will give you:

x+1 > -rac{4}{3}

Notice that we've flipped the inequality sign because we multiplied by a negative number. This is just another way to represent the same condition as -(x+1) < rac{4}{3}, and it often makes the next steps a little easier to visualize.

So, now we have two clear cases:

• Case 1: x+1 < rac{4}{3}
• Case 2: x+1 > -rac{4}{3}

We've successfully transformed our single absolute value inequality into two separate inequalities. This is a huge step forward! Each of these inequalities represents a piece of the puzzle, and by solving them both, we'll find the complete solution set for our original problem. Get ready to put on your solving hats – we're almost there!

Step 3: Solve Each Case

Alright, let's put our solving skills to the test! We have two cases to crack, and each one will give us a piece of the solution. Remember, we're aiming to isolate x in each inequality.

Case 1: x+1 < rac{4}{3}

This one's pretty straightforward. To get x by itself, we simply subtract 1 from both sides of the inequality. This gives us:

x+1 - 1 < rac{4}{3} - 1

Simplifying, we get:

x < rac{4}{3} - rac{3}{3}

x < rac{1}{3}

Fantastic! We've solved for x in Case 1. This tells us that all values of x less than rac{1}{3} are part of the solution.

Case 2: x+1 > -rac{4}{3}

This case is very similar to the first. Again, we want to isolate x, so we subtract 1 from both sides of the inequality:

x+1 - 1 > -rac{4}{3} - 1

Simplifying, we get:

x > -rac{4}{3} - rac{3}{3}

x > -rac{7}{3}

Excellent! We've solved for x in Case 2 as well. This tells us that all values of x greater than -rac{7}{3} are also part of the solution.

Now we have the solutions for both cases:

• Case 1: x < rac{1}{3}
• Case 2: x > -rac{7}{3}

These two inequalities define the range of values for x that satisfy our original absolute value inequality. But we're not quite finished yet! We need to combine these two solutions to get the final answer. Think of it like putting the pieces of a puzzle together – each case gives us a piece, and now we need to see how they fit.

Solving each case individually is like zooming in on different parts of the problem. Now it's time to zoom back out and see the big picture. We've found the boundaries for x, and in the next step, we'll put them together in the correct form. You're doing great – keep going!

Step 4: Combine the Solutions

We've arrived at the final stage! We have the solutions from our two cases:

• x < rac{1}{3}
• x > -rac{7}{3}

Now, we need to combine these into a single compound inequality. A compound inequality is simply a way of expressing that x lies between two values. In our case, x is greater than -rac{7}{3} and less than rac{1}{3}. We can write this as:

-rac{7}{3} < x < rac{1}{3}

Voilà! We've found the solution to our absolute value inequality. This inequality tells us that x can be any number between -rac{7}{3} and rac{1}{3}, not including the endpoints themselves (because of the '<' and '>' signs, not '≤' or '≥').

Remember, the original question asked for the answer in the form $A < x < B$. Comparing this to our solution, -rac{7}{3} < x < rac{1}{3}, we can clearly see that:

A = -rac{7}{3}

B = rac{1}{3}

And there you have it! We've successfully navigated the world of absolute value inequalities. We isolated the absolute value, split the problem into two cases, solved each case individually, and then combined the solutions into a single, elegant answer.

Combining the solutions is like putting the final brushstrokes on a masterpiece. We've taken all the individual pieces and created a complete picture. This final step is where everything comes together, and it's so satisfying to see the solution in its entirety. You've tackled a challenging problem and come out on top – awesome job!

Final Answer

So, to recap, we solved the inequality $3|x+1|+3<7$ and found that the solution is in the form $A < x < B$, where:

A = -rac{7}{3}

B = rac{1}{3}

We did it! We successfully tackled an absolute value inequality. Remember the key steps: isolate the absolute value, split into two cases, solve each case, and combine the solutions. You've now got another tool in your mathematical arsenal. Keep practicing, and you'll become a master of inequalities in no time!

• A = -7/3
• B = 1/3