Solve for \( x \):

\[ |2x - 3| < 9 \]

-9\\\\2x < 9+3\ \wedge\ 2x > -9+3\\\\2x < 12\ \wedge\ 2x > -6\\\\x < 6\ \wedge\ x > -3\\\\\\x\in(-3;\ 6)"> ∣2 x − 3∣ < 9 ⟺ 2 x − 3 < 9 ∧ 2 x − 3 > − 9 2 x < 9 + 3 ∧ 2 x > − 9 + 3 2 x < 12 ∧ 2 x > − 6 x < 6 ∧ x > − 3 x ∈ ( − 3 ; 6 )

Answered by Anonymous | 2024-06-10

-9 \\ \\ 2x < 9 +3 \ \ and \ \ 2x > -9 +3 \\ \\ 2x < 12\ \ and \ \ 2x > -6 \\ \\ x < 6\ \ and \ \ x > - 3 \\ \\ x \in (-3,6)"> ∣2 x − 3∣ < 9 2 x − 3 < 9 an d 2 x − 3 > − 9 2 x < 9 + 3 an d 2 x > − 9 + 3 2 x < 12 an d 2 x > − 6 x < 6 an d x > − 3 x ∈ ( − 3 , 6 )

Answered by Lilith | 2024-06-10

The solution to the inequality ∣2 x − 3∣ < 9 is the interval ( − 3 , 6 ) , meaning that x can take any value between -3 and 6, not including -3 and 6 themselves. This is found by solving two separate inequalities derived from the absolute value inequality. The final solution is expressed as x ∈ ( − 3 , 6 ) .
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Answered by Lilith | 2024-09-30