You can use the addition principle to simplify the problem a bit before proceeding. 3/4(3x - 1/2) - 2/3 < 1/3 3/4(3x - 1/2) - 2/3 + 2/3 < 1/3 + 2/3 (show adding 2/3 to both sides) 3/4(3x - 1/2) < 1 (carry out the addition) We'll use the distributive property to eliminate parentheses at this point. (3/4)(3x) - (3/4)(1/2) < 1 (9/4)x - 3/8 < 1 Now, we can use the multiplication principle to clear fractions. (9/4)x*8 - (3/8)*8 < 1*8 (show multiplication by 8) 18x - 3 < 8 (carry out the multiplication) Use the addition principle again to eliminate the constant term on the left side. 18x - 3 + 3 < 8 + 3 (show addition of 3) 18x < 11 (carry out the addition) Use the multiplication principle to get x by itself. (18x)*(1/18) < 11*(1/18) (show multiplication by 1/18) x < 11/18 (carry out the multiplication)
Comments on the solution process. The steps of the problem would customarily be shown without using a "show" step followed by a "do" step. Usually only the "do" steps would be shown. I have put in the "show" steps here for the purpose of demonstrating the application of the (addition, multiplication) principles you are to use. Often, you are told to "clear fractions first" when solving a problem of this sort. To do that completely in one step requires multiplication by 24, with a factor of 12 applied outside the parentheses on the left and a factor of 2 applied inside. This is sufficiently complicated that I chose to illustrate a different approach. Doing what I said here, you would get 9(6x-1) - 16 < 8 One might chose to "undo" what was done to the variable in this inequality more or less in reverse order of how it was done. To do that here, we would add 16, divide by 9, add 1, divide by 6, and reduce the resulting fraction. The solution above takes a slightly different approach, removing parentheses early. The result is that we use a smaller multiplier to clear fractions and the end result does not need to be reduced. There are many ways to approach a problem of this nature. Pick one that you understand and can do easily.
Comments on the solution process. The steps of the problem would customarily be shown without using a "show" step followed by a "do" step. Usually only the "do" steps would be shown. I have put in the "show" steps here for the purpose of demonstrating the application of the (addition, multiplication) principles you are to use. Often, you are told to "clear fractions first" when solving a problem of this sort. To do that completely in one step requires multiplication by 24, with a factor of 12 applied outside the parentheses on the left and a factor of 2 applied inside. This is sufficiently complicated that I chose to illustrate a different approach. Doing what I said here, you would get 9(6x-1) - 16 < 8 One might chose to "undo" what was done to the variable in this inequality more or less in reverse order of how it was done. To do that here, we would add 16, divide by 9, add 1, divide by 6, and reduce the resulting fraction. The solution above takes a slightly different approach, removing parentheses early. The result is that we use a smaller multiplier to clear fractions and the end result does not need to be reduced. There are many ways to approach a problem of this nature. Pick one that you understand and can do easily.
