Answer :
Certainly! Let's solve each inequality step by step to find all real numbers [tex]\( x \)[/tex] that satisfy the conditions.
### Solving the First Inequality: [tex]\( 18x - 9 < 9 \)[/tex]
1. Add 9 to both sides:
[tex]\[ 18x - 9 + 9 < 9 + 9 \][/tex]
[tex]\[ 18x < 18 \][/tex]
2. Divide both sides by 18:
[tex]\[ x < \frac{18}{18} \][/tex]
[tex]\[ x < 1 \][/tex]
### Solving the Second Inequality: [tex]\( 2x - 4 > -10 \)[/tex]
1. Add 4 to both sides:
[tex]\[ 2x - 4 + 4 > -10 + 4 \][/tex]
[tex]\[ 2x > -6 \][/tex]
2. Divide both sides by 2:
[tex]\[ x > \frac{-6}{2} \][/tex]
[tex]\[ x > -3 \][/tex]
### Combining the Solutions
Now, let's combine the solutions of both inequalities:
- From the first inequality, we have: [tex]\( x < 1 \)[/tex]
- From the second inequality, we have: [tex]\( x > -3 \)[/tex]
When combining the solutions [tex]\( x < 1 \)[/tex] or [tex]\( x > -3 \)[/tex], it indicates that any real number will satisfy at least one of these conditions.
To confirm this, note that [tex]\( x < 1 \)[/tex] covers all numbers less than 1, and [tex]\( x > -3 \)[/tex] covers all numbers greater than [tex]\(-3\)[/tex]. Importantly, any negative number, zero, and positive number up to 1 satisfies these conditions.
Therefore, all real numbers satisfy the inequality.
The correct answer is:
[tex]\[ \text{All real numbers satisfy the inequality.} \][/tex]
### Solving the First Inequality: [tex]\( 18x - 9 < 9 \)[/tex]
1. Add 9 to both sides:
[tex]\[ 18x - 9 + 9 < 9 + 9 \][/tex]
[tex]\[ 18x < 18 \][/tex]
2. Divide both sides by 18:
[tex]\[ x < \frac{18}{18} \][/tex]
[tex]\[ x < 1 \][/tex]
### Solving the Second Inequality: [tex]\( 2x - 4 > -10 \)[/tex]
1. Add 4 to both sides:
[tex]\[ 2x - 4 + 4 > -10 + 4 \][/tex]
[tex]\[ 2x > -6 \][/tex]
2. Divide both sides by 2:
[tex]\[ x > \frac{-6}{2} \][/tex]
[tex]\[ x > -3 \][/tex]
### Combining the Solutions
Now, let's combine the solutions of both inequalities:
- From the first inequality, we have: [tex]\( x < 1 \)[/tex]
- From the second inequality, we have: [tex]\( x > -3 \)[/tex]
When combining the solutions [tex]\( x < 1 \)[/tex] or [tex]\( x > -3 \)[/tex], it indicates that any real number will satisfy at least one of these conditions.
To confirm this, note that [tex]\( x < 1 \)[/tex] covers all numbers less than 1, and [tex]\( x > -3 \)[/tex] covers all numbers greater than [tex]\(-3\)[/tex]. Importantly, any negative number, zero, and positive number up to 1 satisfies these conditions.
Therefore, all real numbers satisfy the inequality.
The correct answer is:
[tex]\[ \text{All real numbers satisfy the inequality.} \][/tex]
