Answer :
To solve the inequality \( 2(19x + 18) > 4 + 11x \), we will proceed step by step to isolate \( x \).
1. Expand the left-hand side of the inequality:
[tex]\[ 2(19x + 18) = 2 \cdot 19x + 2 \cdot 18 = 38x + 36 \][/tex]
Therefore, the inequality becomes:
[tex]\[ 38x + 36 > 4 + 11x \][/tex]
2. Move all terms involving \( x \) to one side and constant terms to the other side:
Subtract \( 11x \) from both sides:
[tex]\[ 38x + 36 - 11x > 4 + 11x - 11x \][/tex]
Simplifies to:
[tex]\[ 27x + 36 > 4 \][/tex]
3. Isolate the term with \( x \):
Subtract 36 from both sides:
[tex]\[ 27x + 36 - 36 > 4 - 36 \][/tex]
Simplifies to:
[tex]\[ 27x > -32 \][/tex]
4. Solve for \( x \) by dividing both sides by 27:
[tex]\[ x > \frac{-32}{27} \][/tex]
Therefore, the solution to the inequality \( 2(19x + 18) > 4 + 11x \) is:
[tex]\[ x > \frac{-32}{27} \][/tex]
In interval notation, this is written as:
[tex]\[ \left( \frac{-32}{27}, \infty \right) \][/tex]
So, the interval notation for the solution is [tex]\(\left( \frac{-32}{27}, \infty \right)\)[/tex].
1. Expand the left-hand side of the inequality:
[tex]\[ 2(19x + 18) = 2 \cdot 19x + 2 \cdot 18 = 38x + 36 \][/tex]
Therefore, the inequality becomes:
[tex]\[ 38x + 36 > 4 + 11x \][/tex]
2. Move all terms involving \( x \) to one side and constant terms to the other side:
Subtract \( 11x \) from both sides:
[tex]\[ 38x + 36 - 11x > 4 + 11x - 11x \][/tex]
Simplifies to:
[tex]\[ 27x + 36 > 4 \][/tex]
3. Isolate the term with \( x \):
Subtract 36 from both sides:
[tex]\[ 27x + 36 - 36 > 4 - 36 \][/tex]
Simplifies to:
[tex]\[ 27x > -32 \][/tex]
4. Solve for \( x \) by dividing both sides by 27:
[tex]\[ x > \frac{-32}{27} \][/tex]
Therefore, the solution to the inequality \( 2(19x + 18) > 4 + 11x \) is:
[tex]\[ x > \frac{-32}{27} \][/tex]
In interval notation, this is written as:
[tex]\[ \left( \frac{-32}{27}, \infty \right) \][/tex]
So, the interval notation for the solution is [tex]\(\left( \frac{-32}{27}, \infty \right)\)[/tex].
