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Graph the solutions to the following inequalities on the number line:

1. [tex]\( x - 99 \leq -104 \)[/tex]
2. [tex]\( x - 51 \leq -43 \)[/tex]
3. [tex]\( 150 + x \leq 144 \)[/tex]
4. [tex]\( x \ \textgreater \ 75 \)[/tex]

Sagot :

GinnyAnswer
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.
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