If it is gien that \( x \) is a nattial number solve for \( x^{2}-3 x \leqslant 4 \) and \( -4 x+3<-2 \)

Solve the inequality by following steps: - step0: Solve for \(x\): \(-4x+3<-2\) - step1: Move the constant to the right side: \(-4x<-2-3\) - step2: Subtract the numbers: \(-4x<-5\) - step3: Change the signs: \(4x>5\) - step4: Divide both sides: \(\frac{4x}{4}>\frac{5}{4}\) - step5: Divide the numbers: \(x>\frac{5}{4}\) Solve the equation \( x^{2}-3 x - 4 \leq 0 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(x^{2}-3x-4\leq 0\) - step1: Rewrite the expression: \(x^{2}-3x-4=0\) - step2: Factor the expression: \(\left(x-4\right)\left(x+1\right)=0\) - step3: Separate into possible cases: \(\begin{align}&x-4=0\\&x+1=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=4\\&x=-1\end{align}\) - step5: Determine the test intervals: \(\begin{align}&x<-1\\&-14\end{align}\) - step6: Choose a value: \(\begin{align}&x_{1}=-2\\&x_{2}=2\\&x_{3}=5\end{align}\) - step7: Test the chosen value: \(\begin{align}&x<-1\textrm{ }\textrm{is not a solution}\\&-14\textrm{ }\textrm{is not a solution}\end{align}\) - step8: Include the critical value: \(\begin{align}&-1\leq x\leq 4\textrm{ }\textrm{is the solution}\end{align}\) - step9: The final solution is \(-1\leq x\leq 4:\) \(-1\leq x\leq 4\) Let's solve the inequalities step by step. 1. **First Inequality: \( x^{2} - 3x \leq 4 \)** We can rearrange this inequality to standard form: \[ x^{2} - 3x - 4 \leq 0 \] Now, we will find the roots of the equation \( x^{2} - 3x - 4 = 0 \) using the quadratic formula. The roots are: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a = 1, b = -3, c = -4 \). The roots are: \[ x = \frac{3 \pm \sqrt{(-3)^{2} - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \] This gives us the roots: \[ x = 4 \quad \text{and} \quad x = -1 \] The inequality \( x^{2} - 3x - 4 \leq 0 \) holds between the roots: \[ -1 \leq x \leq 4 \] 2. **Second Inequality: \( -4x + 3 < -2 \)** Rearranging gives: \[ -4x < -5 \] Dividing by -4 (and flipping the inequality): \[ x > \frac{5}{4} \] 3. **Combining the Results:** We have two conditions: - From the first inequality: \( -1 \leq x \leq 4 \) - From the second inequality: \( x > \frac{5}{4} \) Since \( x \) is a natural number, we need to find the natural numbers that satisfy both conditions. The natural numbers in the range \( -1 \leq x \leq 4 \) are \( 1, 2, 3, 4 \). However, we also need \( x > \frac{5}{4} \), which means \( x \) must be at least \( 2 \). Therefore, the valid natural numbers that satisfy both inequalities are: \[ x = 2, 3, 4 \] Thus, the solution for \( x \) is \( 2, 3, 4 \).