Number Line Solution Set For Inequality -2x + 9 < X - 9

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Hey guys! Today, we're diving into the world of inequalities and number lines. It might sound a bit intimidating, but trust me, it's like solving a puzzle. We've got this inequality: $-2x + 9 < x - 9$, and our mission is to find the number line that perfectly represents its solution set. So, buckle up, grab your thinking caps, and let's get started!

Decoding the Inequality: A Step-by-Step Guide

Before we can match the inequality to a number line, we need to solve it. Think of it as translating a secret code – we need to isolate 'x' to understand what values satisfy the inequality. Here's how we can break it down:

  1. Gather the 'x' terms: Our first move is to bring all the terms with 'x' to one side of the inequality. To do this, we can add $2x$ to both sides. This gives us:

    9<3x−99 < 3x - 9

    Why do we do this? Well, we want to simplify the inequality and make it easier to work with. By grouping the 'x' terms, we're one step closer to isolating 'x'.

  2. Isolate the 'x' term: Now, we need to get rid of the constant term on the right side. We can do this by adding 9 to both sides:

    18<3x18 < 3x

    Remember, whatever we do to one side of the inequality, we must do to the other side to maintain the balance. It's like a seesaw – if we add weight to one side, we need to add the same weight to the other side to keep it level.

  3. Solve for 'x': Finally, to get 'x' all by itself, we need to divide both sides by 3:

    6<x6 < x

    This is a crucial step! We've now isolated 'x', but it's important to remember what this inequality means. It tells us that 'x' is greater than 6. This is different from 'x' being greater than or equal to 6.

  4. Rewriting the inequality (Optional but Recommended): Some people find it easier to read the inequality with 'x' on the left side. We can simply flip the inequality around, but we need to remember to flip the inequality sign as well:

    x>6x > 6

    This might seem like a small change, but it can make a big difference in understanding the solution. Reading it as "x is greater than 6" is often more intuitive.

Key Takeaway: Solving inequalities is all about isolating the variable. Remember to perform the same operations on both sides and be mindful of flipping the inequality sign when multiplying or dividing by a negative number (which we didn't need to do in this case).

Number Line Navigation: Spotting the Correct Solution Set

Okay, so we've cracked the code – our solution is $x > 6$. Now, the fun part begins: matching this solution to the correct number line. Number lines are visual representations of numbers, and they're super helpful for understanding inequalities.

Let's break down what we need to look for on the number line:

  1. The Critical Value: The most important thing is to locate the number 6 on the number line. This is our critical value, the point where the solution begins.

    Think of the critical value as a boundary. It separates the numbers that are solutions to the inequality from those that are not.

  2. The Open Circle: Because our inequality is $x > 6$, and not $x \geq 6$, we need an open circle at 6. This is crucial. An open circle indicates that 6 itself is not included in the solution set. The solution includes all numbers greater than 6, but not 6 itself.

    Imagine the open circle as a fence with a gap. Numbers can get close to 6, but they can't actually touch it.

  3. The Arrow's Direction: Since $x$ is greater than 6, we need the arrow to point to the right. This indicates that all numbers to the right of 6 are solutions to the inequality.

    The arrow shows the direction of the solution. It tells us which way the numbers are getting bigger (or smaller, depending on the inequality).

Analyzing the Options: A Process of Elimination

Now, let's imagine we have four number line options (A, B, C, and D). We can use our knowledge to eliminate the incorrect ones:

  • Eliminate options without 6: Any number line that doesn't have 6 marked is immediately out. We need that critical value!
  • Eliminate options with a closed circle at 6: If a number line has a closed circle (a filled-in circle) at 6, it's incorrect. Remember, our inequality is strictly greater than 6, so 6 is not included.
  • Eliminate options with the arrow pointing the wrong way: If the arrow points to the left, it means the solution is less than 6, which is not what we want. We need the arrow pointing to the right.

By systematically eliminating the wrong options, you'll be left with the correct number line that perfectly represents the solution set $x > 6$.

Pro Tip: When you're unsure, try plugging in a number from the shaded region of the number line into the original inequality. If it makes the inequality true, you're on the right track!

The Importance of Precision: Open vs. Closed Circles

Let's zoom in on a super important detail: the difference between open and closed circles on a number line. This is where many students stumble, so let's make sure we've got it crystal clear.

  • Open Circle (o): An open circle means the critical value is not included in the solution set. This is used for strict inequalities: $<$ (less than) and $>$ (greater than).
  • Closed Circle (●): A closed circle means the critical value is included in the solution set. This is used for inequalities that include