Let's solve the given compound inequality step-by-step.
### Inequality Split
The compound inequality given is:
[tex]\[ 3 \leq -2x + 7 < 5 \][/tex]
We will address each part of the inequality separately and then combine the results.
#### Solving the Left Part: [tex]\( 3 \leq -2x + 7 \)[/tex]
1. Subtract 7 from both sides:
[tex]\[ 3 - 7 \leq -2x + 7 - 7 \][/tex]
[tex]\[ -4 \leq -2x \][/tex]
2. Divide both sides by -2. Remember, when you divide by a negative number, you must reverse the inequality:
[tex]\[ \frac{-4}{-2} \geq \frac{-2x}{-2} \][/tex]
[tex]\[ 2 \geq x \][/tex]
This can also be written as:
[tex]\[ x \leq 2 \][/tex]
#### Solving the Right Part: [tex]\( -2x + 7 < 5 \)[/tex]
1. Subtract 7 from both sides:
[tex]\[ -2x + 7 - 7 < 5 - 7 \][/tex]
[tex]\[ -2x < -2 \][/tex]
2. Divide both sides by -2, reversing the inequality:
[tex]\[ \frac{-2x}{-2} > \frac{-2}{-2} \][/tex]
[tex]\[ x > 1 \][/tex]
### Combining the results
Now we combine the two inequality results:
[tex]\[ x > 1 \quad \text{and} \quad x \leq 2 \][/tex]
This can be written as:
[tex]\[ 1 < x \leq 2 \][/tex]
### Graphing the Solution
To graph this inequality on a number line:
1. Draw an open circle at [tex]\( x = 1 \)[/tex] to indicate that 1 is not included in the solution.
2. Draw a closed circle at [tex]\( x = 2 \)[/tex] to indicate that 2 is included in the solution.
3. Shade the region between 1 and 2.
### Interval Notation
The interval notation for this solution is:
[tex]\[ (1, 2] \][/tex]
### Summary
- Compound Inequality: [tex]\( 3 \leq -2x + 7 < 5 \)[/tex]
- Solution: [tex]\( 1 < x \leq 2 \)[/tex]
- Interval Notation: [tex]\((1, 2]\)[/tex]
