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Sagot :
To solve the inequality [tex]\(|w+3| > 5\)[/tex] and graph the solution on a number line, we break it down into separate cases based on the properties of absolute values.
### Step-by-Step Solution:
1. Understand the absolute value inequality:
[tex]\[ |w + 3| > 5 \][/tex]
This means that the expression [tex]\( w + 3 \)[/tex] is either greater than 5 or less than -5.
2. Set up the two separate inequalities:
[tex]\[ w + 3 > 5 \quad \text{or} \quad w + 3 < -5 \][/tex]
3. Solve each inequality separately:
- For [tex]\( w + 3 > 5 \)[/tex]:
[tex]\[ w + 3 > 5 \][/tex]
Subtract 3 from both sides:
[tex]\[ w > 2 \][/tex]
- For [tex]\( w + 3 < -5 \)[/tex]:
[tex]\[ w + 3 < -5 \][/tex]
Subtract 3 from both sides:
[tex]\[ w < -8 \][/tex]
4. Combine the solutions:
[tex]\[ w > 2 \quad \text{or} \quad w < -8 \][/tex]
In interval notation, this is:
[tex]\[ (-\infty, -8) \cup (2, \infty) \][/tex]
### Graphing the Solution on the Number Line:
To graph this solution on the number line:
- Draw a number line.
- Mark the critical points at [tex]\( -8 \)[/tex] and [tex]\( 2 \)[/tex].
- Use open circles at [tex]\( -8 \)[/tex] and [tex]\( 2 \)[/tex] to indicate that these points are not included in the solution (since the inequality is strict, [tex]\( |w + 3| > 5 \)[/tex] does not include [tex]\( w = -8 \)[/tex] or [tex]\( w = 2 \)[/tex]).
- Shade the region to the left of [tex]\( -8 \)[/tex] (extending to [tex]\( -\infty \)[/tex]) and the region to the right of [tex]\( 2 \)[/tex] (extending to [tex]\( \infty \)[/tex]).
Here's what the number line looks like:
```
<---(===============)----(-8)----(================)--->(2)--(===============)--->
-∞ -8 2 ∞
```
- The section between the open circles [tex]\( -8 \)[/tex] and [tex]\( 2 \)[/tex] is not shaded because these values do not satisfy the inequality.
- The sections extending from [tex]\( -\infty \)[/tex] to [tex]\( -8 \)[/tex] and from [tex]\( 2 \)[/tex] to [tex]\( \infty \)[/tex] are shaded, representing the solution [tex]\( (-\infty, -8) \cup (2, \infty) \)[/tex].
So the solution to the inequality [tex]\(|w + 3| > 5\)[/tex] graphed on the number line includes all [tex]\( w \)[/tex] values less than [tex]\( -8 \)[/tex] and greater than [tex]\( 2 \)[/tex].
### Step-by-Step Solution:
1. Understand the absolute value inequality:
[tex]\[ |w + 3| > 5 \][/tex]
This means that the expression [tex]\( w + 3 \)[/tex] is either greater than 5 or less than -5.
2. Set up the two separate inequalities:
[tex]\[ w + 3 > 5 \quad \text{or} \quad w + 3 < -5 \][/tex]
3. Solve each inequality separately:
- For [tex]\( w + 3 > 5 \)[/tex]:
[tex]\[ w + 3 > 5 \][/tex]
Subtract 3 from both sides:
[tex]\[ w > 2 \][/tex]
- For [tex]\( w + 3 < -5 \)[/tex]:
[tex]\[ w + 3 < -5 \][/tex]
Subtract 3 from both sides:
[tex]\[ w < -8 \][/tex]
4. Combine the solutions:
[tex]\[ w > 2 \quad \text{or} \quad w < -8 \][/tex]
In interval notation, this is:
[tex]\[ (-\infty, -8) \cup (2, \infty) \][/tex]
### Graphing the Solution on the Number Line:
To graph this solution on the number line:
- Draw a number line.
- Mark the critical points at [tex]\( -8 \)[/tex] and [tex]\( 2 \)[/tex].
- Use open circles at [tex]\( -8 \)[/tex] and [tex]\( 2 \)[/tex] to indicate that these points are not included in the solution (since the inequality is strict, [tex]\( |w + 3| > 5 \)[/tex] does not include [tex]\( w = -8 \)[/tex] or [tex]\( w = 2 \)[/tex]).
- Shade the region to the left of [tex]\( -8 \)[/tex] (extending to [tex]\( -\infty \)[/tex]) and the region to the right of [tex]\( 2 \)[/tex] (extending to [tex]\( \infty \)[/tex]).
Here's what the number line looks like:
```
<---(===============)----(-8)----(================)--->(2)--(===============)--->
-∞ -8 2 ∞
```
- The section between the open circles [tex]\( -8 \)[/tex] and [tex]\( 2 \)[/tex] is not shaded because these values do not satisfy the inequality.
- The sections extending from [tex]\( -\infty \)[/tex] to [tex]\( -8 \)[/tex] and from [tex]\( 2 \)[/tex] to [tex]\( \infty \)[/tex] are shaded, representing the solution [tex]\( (-\infty, -8) \cup (2, \infty) \)[/tex].
So the solution to the inequality [tex]\(|w + 3| > 5\)[/tex] graphed on the number line includes all [tex]\( w \)[/tex] values less than [tex]\( -8 \)[/tex] and greater than [tex]\( 2 \)[/tex].
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