It’s no secret that inequalities can seem intimidating at first glance. However, understanding how to represent them on a number line can make all the difference. Take, for example, the inequality –4(x + 3) ≤ –2 – 2x. It might look complex now but by breaking it down step-by-step and plotting it on a number line, we’ll see its simplicity unravel.
First off, let’s tackle what this inequality is asking us to do. In essence, we’re tasked with figuring out which numbers when plugged into ‘x’ make the statement true. That’s our solution set! To visualize these solutions better and understand their scope in relation to each other, we take help from a number line.
The magic of using number lines lies in their ability to provide a clear picture of where our solutions lie within the great expanse of numbers. By representing solutions graphically on such a line, we’re essentially painting an easy-to-understand picture of all possible values for ‘x’ that satisfy our inequality.
Which Number Line Represents The Solution Set For The Inequality –4(x + 3) ≤ –2 – 2x?
Defining the Inequality on a Number Line
Let’s dive straight into understanding inequalities. When we’re dealing with an inequality like -4(x + 3) ≤ –2 – 2x, it may look daunting at first glance. But hold up, it’s not as scary as it seems! Our goal here is to simplify this equation and then plot the solution set on a number line.
So, how do we simplify this inequality? Let me walk you through it. First, distribute the -4 in -4(x + 3). This gives us -4x -12. Now, our inequality looks like this: -4x -12 ≤ –2 – 2x.
Next step? Combine like terms! Move all x terms to one side of the inequality and constants to another. We’ll add 4x to both sides which cancels out ‘–4x’ on left hand side and adds up to ‘2x’ on right hand side giving us ‘–12 ≤ –2 + 6x’.
Lastly, add 2 to both sides ending up with ‘–10 ≤ 6x’. And hey presto! That’s our simplified inequality.
Representation of Solution Set for –4(x + 3) ≤ –2 – 2x
Now that we’ve got our simplified inequality (which reads much easier as ‘6x ≥ -10’), let’s visualize it on a number line. Here’s how:
- Start by isolating x: divide every term by six resulting in ‘ x ≥ -10/6’. Simplify ‘-10/6’ taking into account that negative divided by positive is always negative yielding ‘X ≥ -5/3’.
- Mark ‘-5/3’ or roughly ‘-1.67’ on a number line. Since our inequality includes ‘greater than or equal to’, we’ll use a closed circle at -5/3 to represent that this point is part of the solution set.
- From here, since x can be greater than -5/3, draw an arrow pointing towards the positive end of the number line starting from -5/3. This represents all numbers greater than or equal to -5/3.
Comparing Number Lines to Find the Correct Representation
Now you’ve got your number line representing ‘x ≥ -5/3’. If you’re given multiple number lines, it’s time for comparison! Look for one that has a closed circle at approximately ‘-1.67’ with an arrow pointing towards positive infinity. That’ll be your match!
In summary, breaking down and simplifying our initial inequality –4(x + 3) ≤ –2 – 2x makes it easier to visualize and understand its representation on a number line. Remember, practice makes perfect when it comes to inequalities and their visual representations!
-4(x + 3) ≤ –2 – 2x: What Does This Mean?
Let me take you through the meaning of this inequality. It’s a mathematical statement, just like an equation. The only difference is that instead of equating two expressions, we’re comparing them.
The inequality -4(x + 3) ≤ –2 – 2x can be rewritten as -4x-12 ≤ -2-2x after applying the distributive property. On further simplification, it becomes -4x+2x ≤ -2+12 which leads to -2x ≤10.
That’s where the number line comes in handy. The solution set for this inequality lies on a number line from -5 (inclusive) to positive infinity.
Here’s what the data looks like:
| Inequality | Simplified form | Number Line Start | Number Line End |
| -4(x + 3) ≤ –2 – 2x | -2x≤10 | -5 | ∞ |
Remember:
- When x is multiplied by a negative number, the direction of the inequality changes.
- ‘≤’ means less than or equal to. So when we say x ≥ -5, it includes all numbers greater than or equal to -5.
- Infinity (∞) means there’s no end point on our solution set on the right side of our number line.
In conclusion, any value of x that is greater than or equal to -5 will satisfy our original inequality (-4(x + 3) ≤ –2 – 2x).
