[size=150]Consider the unfinished equation [math]12(x-3)+18=\underscore\underscore\underscore\underscore\underscore\underscore\underscore\underscore\underscore\underscore.[/math][/size][size=150]What is the number of solutions the equation would have with the following expression on the right hand side?[br][br][math]6\left(2x-3\right)[/math][/size]

[math]4(3x-3)[/math]

[math]4(2x-3)[/math]

Describe the defining characteristics of those categories and be prepared to share your reasoning with the class.

[size=150][br][math]6x+8=7x+13[/math][/size]

[math]6x+8=2(3x+4)[/math]

[math]6x+8=6x+13[/math]

If an equation has one solution, solve to find the value of [math]x[/math] that makes the statement true.[br][br][math]6x+8=7x+13[/math][br][math]6x+8=2(3x+4)[/math][br][math]6x+8=6x+13[/math]

[math]\frac{1}{4}(12-4x)=3-x[/math]

[math]x-3=3-x[/math]

[math]x-3=3+x[/math]

If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]\frac{1}{4}(12-4x)=3-x[/math][br][math]x-3=3-x[/math][br][math]x-3=3+x[/math]

[math]-5x-3x+2=-8x+2[/math]

[math]-5x-3x-4=-8x+2[/math]

[math]-5x-4x-2=-8x+2[/math]

If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]-5x-3x+2=-8x+2[/math][br][math]-5x-3x-4=-8x+2[/math][br][math]-5x-4x-2=-8x+2[/math]

[math]4(2x-2)+2=4(x-2)[/math]

[math]4x+2(2x-3)=8(x-1)[/math]

[math]4x+2(2x-3)=4(2x-2)+2[/math]

If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]4(2x-2)+2=4(x-2)[/math][br][math]4x+2(2x-3)=8(x-1)[/math][br][math]4x+2(2x-3)=4(2x-2)+2[/math]

[math]x-3(2-3x)=2(5x+3)[/math]

[math]x-3(2+3x)=2(5x-3)[/math]

[math]x-3(2-3x)=2(5x-3)[/math]

If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]x-3(2-3x)=2(5x+3)[/math][br][math]x-3(2+3x)=2(5x-3)[/math][br][math]x-3(2-3x)=2(5x-3)[/math]

[size=150][size=100]What do you notice about equations with one solution? [/size][/size]

[size=150][size=100]How is this different from equations with no solutions and equations that are true for every [math]x[/math]?[/size][/size]

[size=150][size=100]Choose any set of three consecutive numbers. Find their average. What do you notice?[/size][/size]

[size=150][size=100]Find the average of another set of three consecutive numbers. What do you notice?[/size][/size]

[size=150][size=100]Explain why the thing you noticed must always work, or find a counterexample.[/size][/size]