Answer :
To solve the compound inequality [tex]\( 3x + 1 > 13 \)[/tex] or [tex]\( 5 - 4x < 17 \)[/tex], we will handle each inequality separately and then determine the solution set that satisfies either one or both of the inequalities (since "or" is mentioned, only one condition needs to be true).
### Solving the First Inequality: [tex]\( 3x + 1 > 13 \)[/tex]
1. Subtract 1 from both sides:
[tex]\[ 3x + 1 - 1 > 13 - 1 \][/tex]
[tex]\[ 3x > 12 \][/tex]
2. Divide both sides by 3:
[tex]\[ \frac{3x}{3} > \frac{12}{3} \][/tex]
[tex]\[ x > 4 \][/tex]
### Solving the Second Inequality: [tex]\( 5 - 4x < 17 \)[/tex]
1. Subtract 5 from both sides:
[tex]\[ 5 - 4x - 5 < 17 - 5 \][/tex]
[tex]\[ -4x < 12 \][/tex]
2. Divide both sides by -4. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[ \frac{-4x}{-4} > \frac{12}{-4} \][/tex]
[tex]\[ x > -3 \][/tex]
### Combining the Solution Sets
- From [tex]\( 3x + 1 > 13 \)[/tex], we get [tex]\( x > 4 \)[/tex].
- From [tex]\( 5 - 4x < 17 \)[/tex], we get [tex]\( x > -3 \)[/tex].
Since the overall condition is "or", we combine these two conditions.
Thus, our solution set is the union of the two individual sets. Since [tex]\( x > 4 \)[/tex] encompasses the valid solutions for each inequality (as it is a subset of [tex]\( x > -3 \)[/tex]), we conclude the solution is:
[tex]\[ x > 4 \][/tex]
### Final Answer
The solution to the compound inequality [tex]\( 3x + 1 > 13 \)[/tex] or [tex]\( 5 - 4x < 17 \)[/tex] is:
[tex]\[ x > 4 \][/tex]
### Solving the First Inequality: [tex]\( 3x + 1 > 13 \)[/tex]
1. Subtract 1 from both sides:
[tex]\[ 3x + 1 - 1 > 13 - 1 \][/tex]
[tex]\[ 3x > 12 \][/tex]
2. Divide both sides by 3:
[tex]\[ \frac{3x}{3} > \frac{12}{3} \][/tex]
[tex]\[ x > 4 \][/tex]
### Solving the Second Inequality: [tex]\( 5 - 4x < 17 \)[/tex]
1. Subtract 5 from both sides:
[tex]\[ 5 - 4x - 5 < 17 - 5 \][/tex]
[tex]\[ -4x < 12 \][/tex]
2. Divide both sides by -4. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[ \frac{-4x}{-4} > \frac{12}{-4} \][/tex]
[tex]\[ x > -3 \][/tex]
### Combining the Solution Sets
- From [tex]\( 3x + 1 > 13 \)[/tex], we get [tex]\( x > 4 \)[/tex].
- From [tex]\( 5 - 4x < 17 \)[/tex], we get [tex]\( x > -3 \)[/tex].
Since the overall condition is "or", we combine these two conditions.
Thus, our solution set is the union of the two individual sets. Since [tex]\( x > 4 \)[/tex] encompasses the valid solutions for each inequality (as it is a subset of [tex]\( x > -3 \)[/tex]), we conclude the solution is:
[tex]\[ x > 4 \][/tex]
### Final Answer
The solution to the compound inequality [tex]\( 3x + 1 > 13 \)[/tex] or [tex]\( 5 - 4x < 17 \)[/tex] is:
[tex]\[ x > 4 \][/tex]
