X(x-1) -(x^2-10) <1-6x 3,5-(x-1, 5) <6-4x

To solve these inequalities, we will approach each one step by step. ### Inequality 1: \( x(x-1) - (x^2 - 10) < 1 - 6x \) 1. Expand and simplify both sides: \[ x^2 - x - x^2 + 10 < 1 - 6x \] This simplifies to: \[ -x + 10 < 1 - 6x \] 2. Add \( x \) to both sides: \[ 10 < 1 - 5x \] 3. Now, add \( 5x \) to both sides: \[ 5x + 10 < 1 \] 4. Subtract 10 from both sides: \[ 5x < -9 \] 5. Finally, divide by 5: \[ x < -\frac{9}{5} \] ### Inequality 2: \( 3.5 - (x - 1.5) < 6 - 4x \) 1. Simplify the left side: \[ 3.5 - x + 1.5 < 6 - 4x \] This simplifies to: \[ 5 - x < 6 - 4x \] 2. Add \( 4x \) to both sides: \[ 5 + 3x < 6 \] 3. Now, subtract 5 from both sides: \[ 3x < 1 \] 4. Finally, divide by 3: \[ x < \frac{1}{3} \] ### Summary of Solutions - From the first inequality: \( x < -\frac{9}{5} \) - From the second inequality: \( x < \frac{1}{3} \) The final solution combines both inequalities, giving the critical point \( x < -\frac{9}{5} \) is the more restrictive condition. Hence, the solution set is: \[ x < -\frac{9}{5} \] This implies that any \( x \) satisfying this condition will also satisfy the second inequality.