\( 7 ( x + 4 ) - ^ { \frac { 2 } { 3 } } ( x - 6 ) \leq 2 [ x - 3 ( x + 5 ) ] \)

Solve the equation by following steps: - step0: Solve for \(x\): \(7\left(x+4\right)-\left(x-6\right)^{\frac{2}{3}}=2\left(x-3\left(x+5\right)\right)\) - step1: Subtract the terms: \(7\left(x+4\right)-\left(x-6\right)^{\frac{2}{3}}=2\left(-2x-15\right)\) - step2: Calculate: \(7x+28-\left(x-6\right)^{\frac{2}{3}}=2\left(-2x-15\right)\) - step3: Calculate: \(7x+28-\left(x-6\right)^{\frac{2}{3}}=-4x-30\) - step4: Move the expression to the left side: \(7x+28-\left(x-6\right)^{\frac{2}{3}}-\left(-4x-30\right)=0\) - step5: Calculate: \(11x+58-\left(x-6\right)^{\frac{2}{3}}=0\) - step6: Move the expression to the right-hand side: \(-\left(x-6\right)^{\frac{2}{3}}=-11x-58\) - step7: Divide both sides: \(\left(x-6\right)^{\frac{2}{3}}=11x+58\) - step8: Simplify the equation: \(\left(\left(x-6\right)^{\frac{2}{3}}\right)^{3}=\left(11x+58\right)^{3}\) - step9: Evaluate: \(x^{2}-12x+36=1331x^{3}+21054x^{2}+111012x+58^{3}\) - step10: Move the expression to the left side: \(x^{2}-12x+36-\left(1331x^{3}+21054x^{2}+111012x+58^{3}\right)=0\) - step11: Calculate: \(-21053x^{2}-111024x+36-1331x^{3}-58^{3}=0\) - step12: Simplify: \(-21053x^{2}-111024x+36-58^{3}-1331x^{3}=0\) - step13: Factor the expression: \(-\left(21053x^{2}+111024x-36+58^{3}+1331x^{3}\right)=0\) - step14: Divide both sides: \(21053x^{2}+111024x-36+58^{3}+1331x^{3}=0\) - step15: Calculate: \(x\approx -4.827788\) Solve the equation \( 7 ( x + 4 ) - ( x - 6 )^{ \frac { 2 } { 3 } } > 2 [ x - 3 ( x + 5 ) ] \). Solve the inequality by following steps: - step0: Solve for \(x\): \(7\left(x+4\right)-\left(x-6\right)^{\frac{2}{3}}>2\left(x-3\left(x+5\right)\right)\) - step1: Subtract the terms: \(7\left(x+4\right)-\left(x-6\right)^{\frac{2}{3}}>2\left(-2x-15\right)\) - step2: Calculate: \(7x+28-\left(x-6\right)^{\frac{2}{3}}=2\left(-2x-15\right)\) - step3: Calculate: \(7x+28-\left(x-6\right)^{\frac{2}{3}}=-4x-30\) - step4: Move the expression to the left side: \(7x+28-\left(x-6\right)^{\frac{2}{3}}-\left(-4x-30\right)>0\) - step5: Calculate: \(11x+58-\left(x-6\right)^{\frac{2}{3}}>0\) - step6: Change the signs: \(\left(x-6\right)^{\frac{2}{3}}-11x-58<0\) - step7: Move the expression to the right side: \(\left(x-6\right)^{\frac{2}{3}}<11x+58\) - step8: Separate into possible cases: \(\begin{align}&\left(x-6\right)^{\frac{2}{3}}<11x+58,11x+58\geq 0\\&\left(x-6\right)^{\frac{2}{3}}<11x+58,11x+58<0\end{align}\) - step9: Solve the inequality: \(\begin{align}&x>-4.827788,x\geq -\frac{58}{11}\\&x \in \varnothing ,x<-\frac{58}{11}\end{align}\) - step10: Find the intersection: \(\begin{align}&x>-4.827788\\&x \in \varnothing \end{align}\) - step11: Find the union: \(x>-4.827788\) To solve the inequality \[ 7 ( x + 4 ) - ( x - 6 )^{ \frac { 2 } { 3 } } \leq 2 [ x - 3 ( x + 5 ) ], \] we first rewrite it as an equation to find the boundary points: \[ 7 ( x + 4 ) - ( x - 6 )^{ \frac { 2 } { 3 } } = 2 [ x - 3 ( x + 5 ) ]. \] After solving this equation, we find one solution: \[ x \approx -4.827788. \] Next, we analyze the inequality: \[ 7 ( x + 4 ) - ( x - 6 )^{ \frac { 2 } { 3 } } > 2 [ x - 3 ( x + 5 ) ]. \] The solution to this inequality gives us: \[ x > -4.827788. \] ### Conclusion: The solution to the original inequality is: \[ x \geq -4.827788. \] This means that \( x \) can take any value greater than or equal to approximately \(-4.83\).