Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Let's solve the inequality [tex]\(-4(x + 3) \leq -2 - 2x\)[/tex] step-by-step to determine the solution set and represent it on a number line.
### Step 1: Distribute [tex]\(-4\)[/tex] on the left side of the inequality
The given inequality is:
[tex]\[ -4(x + 3) \leq -2 - 2x \][/tex]
Distribute [tex]\(-4\)[/tex] inside the parenthesis:
[tex]\[ -4x - 12 \leq -2 - 2x \][/tex]
### Step 2: Add [tex]\(4x\)[/tex] to both sides to combine like terms
To isolate the term with [tex]\(x\)[/tex] on one side, add [tex]\(4x\)[/tex] to both sides:
[tex]\[ -4x - 12 + 4x \leq -2 - 2x + 4x \][/tex]
Simplify the equation:
[tex]\[ -12 \leq -2 + 2x \][/tex]
### Step 3: Add [tex]\(2\)[/tex] to both sides to further isolate the term with [tex]\(x\)[/tex]
Add [tex]\(2\)[/tex] to both sides:
[tex]\[ -12 + 2 \leq 2x \][/tex]
Simplify the equation:
[tex]\[ -10 \leq 2x \][/tex]
### Step 4: Divide both sides by [tex]\(2\)[/tex] to solve for [tex]\(x\)[/tex]
Divide by [tex]\(2\)[/tex]:
[tex]\[ \frac{-10}{2} \leq x \][/tex]
Simplify:
[tex]\[ -5 \leq x \][/tex]
This inequality can be written as:
[tex]\[ x \geq -5 \][/tex]
### Step 5: Represent the solution on the number line
The solution [tex]\(x \geq -5\)[/tex] implies that [tex]\( x \)[/tex] is any number greater than or equal to [tex]\(-5\)[/tex].
To represent this solution on the number line:
1. Draw a number line.
2. Locate the point [tex]\(-5\)[/tex] on the number line.
3. Draw a closed circle at [tex]\(-5\)[/tex] (because [tex]\(-5\)[/tex] is included in the solution set).
4. Shade the region to the right of [tex]\(-5\)[/tex] to indicate all numbers greater than or equal to [tex]\(-5\)[/tex].
The final number line representation for the solution set [tex]\(x \geq -5\)[/tex] looks like this:
```
<----|----|----|----|----|----|----|----|----|--->
-7 -6 -5 -4 -3 -2 -1 0 1
[=====================>
```
The closed circle at [tex]\(-5\)[/tex] and the shading to the right of [tex]\(-5\)[/tex] represent all values [tex]\(x\)[/tex] that are greater than or equal to [tex]\(-5\)[/tex].
### Step 1: Distribute [tex]\(-4\)[/tex] on the left side of the inequality
The given inequality is:
[tex]\[ -4(x + 3) \leq -2 - 2x \][/tex]
Distribute [tex]\(-4\)[/tex] inside the parenthesis:
[tex]\[ -4x - 12 \leq -2 - 2x \][/tex]
### Step 2: Add [tex]\(4x\)[/tex] to both sides to combine like terms
To isolate the term with [tex]\(x\)[/tex] on one side, add [tex]\(4x\)[/tex] to both sides:
[tex]\[ -4x - 12 + 4x \leq -2 - 2x + 4x \][/tex]
Simplify the equation:
[tex]\[ -12 \leq -2 + 2x \][/tex]
### Step 3: Add [tex]\(2\)[/tex] to both sides to further isolate the term with [tex]\(x\)[/tex]
Add [tex]\(2\)[/tex] to both sides:
[tex]\[ -12 + 2 \leq 2x \][/tex]
Simplify the equation:
[tex]\[ -10 \leq 2x \][/tex]
### Step 4: Divide both sides by [tex]\(2\)[/tex] to solve for [tex]\(x\)[/tex]
Divide by [tex]\(2\)[/tex]:
[tex]\[ \frac{-10}{2} \leq x \][/tex]
Simplify:
[tex]\[ -5 \leq x \][/tex]
This inequality can be written as:
[tex]\[ x \geq -5 \][/tex]
### Step 5: Represent the solution on the number line
The solution [tex]\(x \geq -5\)[/tex] implies that [tex]\( x \)[/tex] is any number greater than or equal to [tex]\(-5\)[/tex].
To represent this solution on the number line:
1. Draw a number line.
2. Locate the point [tex]\(-5\)[/tex] on the number line.
3. Draw a closed circle at [tex]\(-5\)[/tex] (because [tex]\(-5\)[/tex] is included in the solution set).
4. Shade the region to the right of [tex]\(-5\)[/tex] to indicate all numbers greater than or equal to [tex]\(-5\)[/tex].
The final number line representation for the solution set [tex]\(x \geq -5\)[/tex] looks like this:
```
<----|----|----|----|----|----|----|----|----|--->
-7 -6 -5 -4 -3 -2 -1 0 1
[=====================>
```
The closed circle at [tex]\(-5\)[/tex] and the shading to the right of [tex]\(-5\)[/tex] represent all values [tex]\(x\)[/tex] that are greater than or equal to [tex]\(-5\)[/tex].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
