Solving Inequalities: Find The Interval A

Hey guys! Today, we're diving into the world of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have specific solutions, inequalities often have a range of solutions. Let's tackle the inequality $7 + 3x < 2x + 4$ and find the interval that represents its solution. Grab your pencils, and let’s get started!

Understanding Inequalities

Before we jump into solving, let's quickly recap what inequalities are all about. Think of an inequality as a balancing scale, but instead of being perfectly balanced, one side is either heavier or lighter than the other. For instance, $a < b$ means 'a' is less than 'b,' while $a > b$ means 'a' is greater than 'b.' These symbols help us express relationships between values that aren't necessarily equal.

When solving inequalities, the goal is to isolate the variable (usually 'x') on one side, just like with equations. However, there's one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers. For example, if $2 < 4$, multiplying both sides by -1 gives $-2 > -4$.

Understanding this rule is essential to avoid making mistakes. Also, remember that the solution to an inequality is often a range of values, which we express using intervals. Intervals are sets of numbers between two endpoints. For example, the interval $(a, b)$ represents all numbers between 'a' and 'b,' not including 'a' and 'b.' The interval $[a, b]$ includes 'a' and 'b.' We use parentheses when the endpoint is not included and brackets when it is included. This stuff is super important, so make sure to keep it in mind!

Solving the Inequality $7 + 3x < 2x + 4$

Alright, let's break down the given inequality step by step. Our inequality is $7 + 3x < 2x + 4$. Our goal is to isolate 'x' on one side. Here's how we can do it:

1. Subtract 2x from both sides: This moves the 'x' terms to one side. So, we have: $7 + 3x - 2x < 2x + 4 - 2x$ Which simplifies to: $7 + x < 4$

2. Subtract 7 from both sides: This isolates 'x' on the left side. Now we get: $7 + x - 7 < 4 - 7$ Which simplifies to: $x < -3$

So, the solution to the inequality is $x < -3$. This means that any value of 'x' that is less than -3 will satisfy the original inequality. Isn't that neat? We've successfully found the solution! It's all about those algebraic manipulations, one step at a time. Remember to keep your eye on the prize: isolating that variable!

Expressing the Solution as an Interval

Now that we've solved the inequality and found that $x < -3$, we need to express this solution as an interval. An interval is a range of values that 'x' can take. Since 'x' is less than -3, it can be any number from negative infinity up to, but not including, -3.

In interval notation, we represent this as $(-\infty, -3)$. The parenthesis next to $-\infty$ indicates that negative infinity is not a specific number, so it can't be included in the interval. The parenthesis next to -3 indicates that -3 is not included in the solution; 'x' must be strictly less than -3.

So, the interval that represents the solution to the inequality $7 + 3x < 2x + 4$ is $(-\infty, -3)$. This is a really clear and concise way to show all the possible values of 'x' that make the inequality true. Interval notation is super handy once you get the hang of it.

Determining the Value of A

The question asks us to find the value of A, where the solution is given in the form $(-\infty, A)$. From our solution, we know that the interval is $(-\infty, -3)$. Therefore, the value of A is -3.

So, $A = -3$. That wasn't too hard, was it? Finding 'A' is just about matching up the interval notation with the solution you've already worked out. Always double-check your work to make sure everything lines up perfectly.

Is the Interval Included in the Solution?

The question also asks whether the interval $(-\infty, A)$ is included in the solution. Since our solution is $x < -3$, the interval representing this solution is $(-\infty, -3)$. The question is a bit redundant because the interval $(-\infty, A)$ is by definition the solution we found.

• Interval $(-\infty, A)$: YES, this interval is included in the solution because it is the solution. This interval represents all values less than A (which is -3), which is exactly what our inequality tells us.

Basically, the question is asking if the solution we found matches the solution we found, which, of course, it does! Sometimes math problems can be a little bit circular like that. Always read the question carefully to make sure you understand what's being asked.

Conclusion

To wrap it up, we started with the inequality $7 + 3x < 2x + 4$, solved it to find $x < -3$, and then expressed the solution as an interval $(-\infty, -3)$. We identified that A = -3 and confirmed that the interval $(-\infty, A)$ is indeed the solution to the inequality.

Inequalities might seem tricky at first, but with practice, you'll become a pro at solving them. Remember to pay attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. Keep practicing, and you'll master inequalities in no time! Keep up the awesome work, guys! And remember, math is like building with LEGOs – each piece fits together to create something amazing!