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Sagot :
To solve the inequalities and represent their solutions on the number line, follow these steps:
### 1. Solve the first inequality: [tex]\( x - 99 \leq -104 \)[/tex]
Add 99 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x - 99 + 99 \leq -104 + 99 \][/tex]
[tex]\[ x \leq -5 \][/tex]
So, the solution to the first inequality is [tex]\( x \leq -5 \)[/tex].
### 2. Solve the second inequality: [tex]\( x - 51 \leq -43 \)[/tex]
Add 51 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x - 51 + 51 \leq -43 + 51 \][/tex]
[tex]\[ x \leq 8 \][/tex]
So, the solution to the second inequality is [tex]\( x \leq 8 \)[/tex].
### 3. Solve the third inequality: [tex]\( 150 + x \leq 144 \)[/tex]
Subtract 150 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ 150 + x - 150 \leq 144 - 150 \][/tex]
[tex]\[ x \leq -6 \][/tex]
So, the solution to the third inequality is [tex]\( x \leq -6 \)[/tex].
### 4. Solve the fourth inequality: [tex]\( 75 < x \)[/tex]
This inequality can be written in standard mathematical notation as:
[tex]\[ x > 75 \][/tex]
So, the solution to the fourth inequality is [tex]\( x > 75 \)[/tex].
### Representing on the Number Line
To represent these solutions on a number line:
1. [tex]\( x \leq -5 \)[/tex]: This includes all numbers to the left of and including -5. Draw a solid circle at -5 and shade everything to the left.
2. [tex]\( x \leq 8 \)[/tex]: This includes all numbers to the left of and including 8. Draw a solid circle at 8 and shade everything to the left.
3. [tex]\( x \leq -6 \)[/tex]: This includes all numbers to the left of and including -6. Draw a solid circle at -6 and shade everything to the left.
4. [tex]\( x > 75 \)[/tex]: This includes all numbers to the right of 75. Draw an open circle at 75 and shade everything to the right.
### Combined Solution
When considering the intersection of these inequalities:
- The solution [tex]\( x \leq -5 \)[/tex] implies that [tex]\( x \)[/tex] must be no greater than -5.
- The solution [tex]\( x \leq -6 \)[/tex] is more restrictive than [tex]\( x \leq -5 \)[/tex], since -6 is less than -5.
The most restrictive [tex]\( x \leq -6 \)[/tex] will take precedence in the combined solution for values less than -5.
- The solution [tex]\( x > 75 \)[/tex] does not intersect with any of the previous solutions, so it stands separately.
Final Representation:
1. [tex]\( x \leq -6 \)[/tex] is one range.
2. [tex]\( x > 75 \)[/tex] is another distinct range.
On the number line:
- Mark a solid circle at -6 and shade everything to the left.
- Mark an open circle at 75 and shade everything to the right.
### 1. Solve the first inequality: [tex]\( x - 99 \leq -104 \)[/tex]
Add 99 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x - 99 + 99 \leq -104 + 99 \][/tex]
[tex]\[ x \leq -5 \][/tex]
So, the solution to the first inequality is [tex]\( x \leq -5 \)[/tex].
### 2. Solve the second inequality: [tex]\( x - 51 \leq -43 \)[/tex]
Add 51 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x - 51 + 51 \leq -43 + 51 \][/tex]
[tex]\[ x \leq 8 \][/tex]
So, the solution to the second inequality is [tex]\( x \leq 8 \)[/tex].
### 3. Solve the third inequality: [tex]\( 150 + x \leq 144 \)[/tex]
Subtract 150 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ 150 + x - 150 \leq 144 - 150 \][/tex]
[tex]\[ x \leq -6 \][/tex]
So, the solution to the third inequality is [tex]\( x \leq -6 \)[/tex].
### 4. Solve the fourth inequality: [tex]\( 75 < x \)[/tex]
This inequality can be written in standard mathematical notation as:
[tex]\[ x > 75 \][/tex]
So, the solution to the fourth inequality is [tex]\( x > 75 \)[/tex].
### Representing on the Number Line
To represent these solutions on a number line:
1. [tex]\( x \leq -5 \)[/tex]: This includes all numbers to the left of and including -5. Draw a solid circle at -5 and shade everything to the left.
2. [tex]\( x \leq 8 \)[/tex]: This includes all numbers to the left of and including 8. Draw a solid circle at 8 and shade everything to the left.
3. [tex]\( x \leq -6 \)[/tex]: This includes all numbers to the left of and including -6. Draw a solid circle at -6 and shade everything to the left.
4. [tex]\( x > 75 \)[/tex]: This includes all numbers to the right of 75. Draw an open circle at 75 and shade everything to the right.
### Combined Solution
When considering the intersection of these inequalities:
- The solution [tex]\( x \leq -5 \)[/tex] implies that [tex]\( x \)[/tex] must be no greater than -5.
- The solution [tex]\( x \leq -6 \)[/tex] is more restrictive than [tex]\( x \leq -5 \)[/tex], since -6 is less than -5.
The most restrictive [tex]\( x \leq -6 \)[/tex] will take precedence in the combined solution for values less than -5.
- The solution [tex]\( x > 75 \)[/tex] does not intersect with any of the previous solutions, so it stands separately.
Final Representation:
1. [tex]\( x \leq -6 \)[/tex] is one range.
2. [tex]\( x > 75 \)[/tex] is another distinct range.
On the number line:
- Mark a solid circle at -6 and shade everything to the left.
- Mark an open circle at 75 and shade everything to the right.
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