All, Some, or No Solutions: IM 8.4.7
[url=https://im.kendallhunt.com/MS/teachers/3/4/7/preparation.html]“All, Some, or No Solutions”[/url] from IM Grade 8 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 8.4.7
- IM 8.4.7 Lesson: All, Some, or No Solutions
- Practice 8.4.7
- IM 8.4.7 Practice: All, Some, or No Solutions
IM 8.4.7 Lesson: All, Some, or No Solutions
Which one doesn’t belong?
[list=1][*][math]5+7=7+5[/math] [/*][*][math]5\cdot7=7\cdot5[/math][/*][*][math]2=7-5[/math][br][/*][*][math]5-7=7-5[/math][br][/*][/list][br]Explain your reasoning.
Select all the equations that are true for [i]all[/i] values.
Select all the equations that are true for [i]no[/i] values.[br]
Write the other side of this equation so that this equation is true for all values of [math]u[/math].[br][br][math]6(u-2)+2=[/math]
Write the other side of this equation so that this equation is true for no values of [math]u[/math]. [br][br][math]6(u-2)+2=[/math]
Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98.
[size=150]How many sets of two or more consecutive positive integers can be added to obtain a sum of 100?[/size]
Complete each equation so that it is true for all values of x.
[math]3x+6=3(x+\underscore\underscore)[/math]
[math]x-2=-(\underscore\underscore-x)[/math]
[math]\frac{15x-10}{5}=\underscore\underscore-2[/math]
Complete each equation so that it is true for no values of x.
[math]3x+6=3(x+\underscore\underscore)[/math]
[math]x-2=-(\underscore\underscore-x)[/math]
[math]\frac{15x-10}{5}=\underscore\underscore-2[/math]
[size=150]Describe how you know whether an equation will be true for all values of [math]x[/math] or true for no values of [math]x[/math].[/size]
IM 8.4.7 Practice: All, Some, or No Solutions
For each equation, decide if it is always true or never true.
[math]x-13=x+1[/math]
[math]x+\frac{1}{2}=x-\frac{1}{2}[/math]
[math]2(x+3)=5x+6-3x[/math]
[math]x-3=2x-3-x[/math]
[math]3(x-5)=2(x-5)+x[/math]
[size=150]Mai says that the equation [math]2x+2=x+1[/math] has no solution because the left hand side is double the right hand side. Do you agree with Mai? Explain your reasoning.[/size]
Write the other side of this equation so it's true for all values of x:
[math]\frac{1}{2}(6x-10)-x=[/math]
Write the other side of this equation so it's true for no values of x:
[math]\frac{1}{2}(6x-10)-x=[/math]
[size=150][size=100]Here is an equation that is true for all values of [math]x[/math]: [math]5(x+2)=5x+10[/math]. Elena saw this equation and says she can tell [math]20(x+2)+31=4(5x+10)+31[/math] is also true for any value of [math]x[/math]. How can she tell? Explain your reasoning.[/size][/size]
[size=150]Elena and Lin are trying to solve [math]\frac{1}{2}x+3=\frac{7}{2}x+5[/math]. Describe the change they each make to each side of the equation.[br][br][size=100]Elena’s first step is to write [math]3=\frac{7}{2}x-\frac{1}{2}x+5[/math][/size][/size].
Lin’s first step is to write[math]x+6=7x+10.[/math]
Solve each equation and check your solution.
[math]3x-6=4(2-3x)-8x[/math]
[math]\frac{1}{2}x+6=\frac{3}{2}(z+6)[/math]
[math]9-7w=8w+8[/math]
[size=100][size=150]The point [math](-3,6)[/math] is on a line with a slope of 4.[/size][br][br]Find two more points on the line.[/size]
Write an equation for the line.