Solve Inequality & Simul Eqns: Alg Workings Q (a,b,c)

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The discussion focuses on solving an inequality and simultaneous equations involving algebraic expressions. Part (a) requires solving the inequality \(5(3x + 1) < 11x\) through standard algebraic methods, emphasizing the importance of inequality direction when multiplying or dividing by negative values. Part (b) involves solving the simultaneous equations \(3x^2 + y^2 - 7 = 0\) and \(y - 3x - 5 = 0\) by substituting \(y\) into the first equation to find \(x\). Finally, part (c) asks for the values of \(x\) that satisfy both the inequality and the equations from part (b).

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Q.
(a) Solve the inequality 5(3x + 1) &lt;11x, Show clear algebraic working. (2)
(b) Solve the simultaneous equations 3x 2 + y 2 – 7 = 0, y – 3x – 5 = 0 Show clear algebraic working. (4)
(c) Hence find the value of x for which 5(x + 1) &lt; x and 3x 2 + y 2 – 7 = 0 and y – 3x – 5 = 0
 
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assuming you mean ...

(a) $5(3x+1) \le 11x$

solve for $x$ as you would any other equation ...
though probably not necessary to know in this case, be aware that multiplying or dividing by a negative value changes the direction of the inequality

(b) $3x^2 + y^2 - 7 = 0$
$y - 3x - 5 = 0 \implies y = 3x+5$

I would substitute $(3x+5)$ for $y$ in the first equation and solve the resulting quadratic for $x$ ... note you'll get two ordered pairs that satisfy the system

(c) Decide which of the two (or maybe both) values for $x$ from part (b) satisfy the given inequality
 

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