Questions: Solve. Write the solution set in set-builder notation. 6-4 x<18 and 1-3 x>-16

Solution Steps

To solve the system of inequalities, we need to solve each inequality separately and then find the intersection of the solution sets. For the first inequality, isolate \(x\) by performing algebraic operations. Do the same for the second inequality. Finally, combine the solutions to find the values of \(x\) that satisfy both inequalities.

We start with the first inequality: \[ 6 - 4x < 18 \] Subtracting 6 from both sides gives: \[ -4x < 12 \] Dividing both sides by -4 (and reversing the inequality) results in: \[ x > -3 \] Thus, the solution for the first inequality is: \[ (-3 < x) \]

Next, we solve the second inequality: \[ 1 - 3x > -16 \] Subtracting 1 from both sides yields: \[ -3x > -17 \] Dividing both sides by -3 (and reversing the inequality) gives: \[ x < \frac{17}{3} \] Therefore, the solution for the second inequality is: \[ (x < \frac{17}{3}) \]

Now, we combine the solutions from both inequalities: \[ -3 < x < \frac{17}{3} \] This represents the intersection of the two solution sets.

Final Answer