Given: $$ 4-\angle x

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Given: $$ 4-\angle x<16 $$ where $$ x \in \AA $$ 2.1.1 Solve the inequality. 2.12 Hence, represent your answer to QUESTION 2.1.1 on a number line. Solve simultaneously for $$ x $$ and $$ y $$ : $$ -2 x-y=10 \text { and } 3 x-4 y=-4 $$ Solve for $$ x $$ : $$ 2.3 .1 \quad \frac{x(x-5)}{6}-1=0 $$ $$ 232 \quad e=\sqrt{a+2 x} $$ Tabelo is currently four times as old as his daughter, Linda. Six years from now, Tahelo will be throc times as old as Linda. Colculate Linda'r age currently.

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Solve the equation by following steps: - step0: Solve for $$x$$: $$\frac{x\left(x-5\right)}{6}-1=0$$ - step1: Multiply both sides of the equation by LCD: $$\left(\frac{x\left(x-5\right)}{6}-1\right)\times 6=0\times 6$$ - step2: Simplify the equation: $$x^{2}-5x-6=0$$ - step3: Factor the expression: $$\left(x-6\right)\left(x+1\right)=0$$ - step4: Separate into possible cases: $$\begin{align}&x-6=0\&x+1=0\end{align}$$ - step5: Solve the equation: $$\begin{align}&x=6\&x=-1\end{align}$$ - step6: Rewrite: $$x_{1}=-1,x_{2}=6$$ Solve the equation $$ e=\sqrt{a+2x} $$. Solve the equation by following steps: - step0: Solve for $$x$$: $$e=\sqrt{a+2x}$$ - step1: Swap the sides: $$\sqrt{a+2x}=e$$ - step2: Raise both sides to the $$2$$-th power$$:$$ $$\left(\sqrt{a+2x}\right)^{2}=e^{2}$$ - step3: Evaluate the power: $$a+2x=e^{2}$$ - step4: Move the expression to the right side: $$2x=e^{2}-a$$ - step5: Divide both sides: $$\frac{2x}{2}=\frac{e^{2}-a}{2}$$ - step6: Divide the numbers: $$x=\frac{e^{2}-a}{2}$$ Solve the system of equations $$ -2x-y=10;3x-4y=-4 $$. Solve the system of equations by following steps: - step0: Solve using the substitution method: $$\left\{ \begin{array}{l}-2x-y=10\3x-4y=-4\end{array}\right.$$ - step1: Solve the equation: $$\left\{ \begin{array}{l}y=-10-2x\3x-4y=-4\end{array}\right.$$ - step2: Substitute the value of $$y:$$ $$3x-4\left(-10-2x\right)=-4$$ - step3: Simplify: $$11x+40=-4$$ - step4: Move the constant to the right side: $$11x=-4-40$$ - step5: Subtract the numbers: $$11x=-44$$ - step6: Divide both sides: $$\frac{11x}{11}=\frac{-44}{11}$$ - step7: Divide the numbers: $$x=-4$$ - step8: Substitute the value of $$x:$$ $$y=-10-2\left(-4\right)$$ - step9: Calculate: $$y=-2$$ - step10: Calculate: $$\left\{ \begin{array}{l}x=-4\y=-2\end{array}\right.$$ - step11: Check the solution: $$\left\{ \begin{array}{l}x=-4\y=-2\end{array}\right.$$ - step12: Rewrite: $$\left(x,y\right) = \left(-4,-2\right)$$ Solve the equation $$ 4-x<16 $$. Solve the inequality by following steps: - step0: Solve for $$x$$: $$4-x<16$$ - step1: Move the constant to the right side: $$-x<16-4$$ - step2: Subtract the numbers: $$-x<12$$ - step3: Change the signs: $$x>-12$$ Solve the system of equations $$ T=4L;T+6=3(L+6) $$. Solve the system of equations by following steps: - step0: Solve using the substitution method: $$\left\{ \begin{array}{l}T=4L\T+6=3\left(L+6\right)\end{array}\right.$$ - step1: Substitute the value of $$T:$$ $$4L+6=3\left(L+6\right)$$ - step2: Expand the expression: $$4L+6=3L+18$$ - step3: Move the expression to the left side: $$4L-3L=18-6$$ - step4: Add and subtract: $$L=18-6$$ - step5: Add and subtract: $$L=12$$ - step6: Substitute the value of $$L:$$ $$T=4\times 12$$ - step7: Calculate: $$T=48$$ - step8: Calculate: $$\left\{ \begin{array}{l}L=12\T=48\end{array}\right.$$ - step9: Check the solution: $$\left\{ \begin{array}{l}L=12\T=48\end{array}\right.$$ - step10: Rewrite: $$\left(L,T\right) = \left(12,48\right)$$ Let's solve the problems step by step. ### 1. Solve the inequality $$ 4 - x < 16 $$ To solve the inequality: $$ 4 - x < 16 $$ Subtract 4 from both sides: $$ -x < 12 $$ Now, multiply both sides by -1 (remember to reverse the inequality sign): $$ x > -12 $$ ### 2. Represent the solution on a number line The solution $$ x > -12 $$ means that $$ x $$ can take any value greater than -12. On a number line, you would represent this by drawing an open circle at -12 (indicating that -12 is not included) and shading to the right. ### 3. Solve the system of equations: $$ -2x - y = 10 \quad \text{and} \quad 3x - 4y = -4 $$ The solution to this system is: $$ (x, y) = (-4, -2) $$ ### 4. Solve for $$ x $$ in the equation: $$ \frac{x(x-5)}{6} - 1 = 0 $$ The solutions are: $$ x_1 = -1 \quad \text{and} \quad x_2 = 6 $$ ### 5. Solve for $$ x $$ in the equation: $$ e = \sqrt{a + 2x} $$ Rearranging gives: $$ x = \frac{e^2 - a}{2} $$ ### 6. Calculate Linda's current age Let $$ T $$ be Tahelo's current age and $$ L $$ be Linda's current age. We have the following equations based on the problem statement: 1. $$ T = 4L $$ (Tahelo is currently four times as old as Linda) 2. $$ T + 6 = 3(L + 6) $$ (In six years, Tahelo will be three times as old as Linda) Substituting the first equation into the second: $$ 4L + 6 = 3(L + 6) $$ Expanding gives: $$ 4L + 6 = 3L + 18 $$ Subtracting $$ 3L $$ from both sides: $$ L + 6 = 18 $$ Subtracting 6 from both sides: $$ L = 12 $$ Thus, Linda's current age is **12 years**. ### Summary of Results: - Inequality solution: $$ x > -12 $$ - System of equations solution: $$ (x, y) = (-4, -2) $$ - $$ x $$ solutions: $$ x_1 = -1, x_2 = 6 $$ - Linda's current age: **12 years**. - The solution to the inequality $$ 4 - x < 16 $$ is $$ x > -12 $$. - The system of equations $$ -2x - y = 10 $$ and $$ 3x - 4y = -4 $$ has the solution $$ (x, y) = (-4, -2) $$. - The solutions to $$ \frac{x(x-5)}{6} - 1 = 0 $$ are $$ x = -1 $$ and $$ x = 6 $$. - The solution to $$ e = \sqrt{a + 2x} $$ is $$ x = \frac{e^2 - a}{2} $$. - Linda is currently **12 years old**.

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