How do you solve this equation: $2x+8=6x-12$ by using the guess and check method?
I divide $2x+8$ and I get $4$ then I divide $6x-12$ and I get $-2$ but I don't know what to do next or is it wrong?
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$\begingroup$ Is it $2^x$, instead of $2x$? For $2x$ there isn't much to guess. $\endgroup$ – Karolis Juodelė Jun 24 '14 at 17:11
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3$\begingroup$ Try guessing $x = 5$. $\endgroup$ – JimmyK4542 Jun 24 '14 at 17:12
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The answer is $x = 5$. You can get this solution using normal analytical methods (algebraic manipulation), i.e., $$\begin{align} 2x + 8 = 6x - 12 &\iff -4x + 8 = -12 \\&\iff -4x = -20 \\&\iff x = 5 \,\,. \end{align}$$ In terms of a "guess and check method", here's my strategy: factor $2$ out of the LHS and $6$ out of the RHS to obtain the equation $2(x + 4) = 6(x - 2)$. Now, we can divide $2$ from both sides to obtain $x + 4 = 3(x - 2)$. The solution is fairly easy to see in this form by guessing and checking. You will come to $x = 5$, as before.
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Your instructor wants you to try plugging in a bunch of numbers to guess the correct value. Just by eyeballing this, we might try 4 or 5. If you plug in $5$ on the left you get $$2(5)+8=18$$ and on the right you get $$6(5)-12 = 30 - 12 = 18.$$ This tells us that $x=5$ is a solution to this equation.
Remember that the old methods still work here. So if you add 12 to both sides and subtract $2x$ from both sides you get $$20=4x$$ and then you can solve for $x$ by dividing both sides by 4.
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One thing that you can get by 'guess-and-check' is how linear functions grow.
$$\begin{array}{r|r|r} x &2x+8 &6x-12\\ \hline 0&8&-12\\1&10&-6\\2&12&0 \\ \vdots&\vdots&\vdots \end{array}$$ Note that every time $x$ increases by $1$, $2x+8$ increases by $2$ and $6x-12$ increases by $6$. (do you see why this should be so?) That means whenever $x$ increases by $1$, $6x-12$ grows by $4$ units more than $2x+8$. Currently (at $x=2$), $2x+8$ is ahead by $12$. So if we increase $x$ by $3$, $6x-12$ will gain $3\cdot 4$, or $12$ more than $2x-8$, and so then the two expressions will be equal.
Thus the answer should be $x=5$.
