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how do i solve 1-3(X+1)+7(X-6)>0

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To solve this inequality, you first need to simplify the left side. 

You can do this by first applying the distributive property; you will distribute the coefficients outside of the parentheses to the terms inside the parentheses through multiplication. 

For the first set of parentheses you multiply -3 by X and by 1. Your result will be -3x and -3. Since you have distributed, you can eliminate the parentheses. 

Your inequality now reads 1-3x -3 +7(x-6)>0

Let's use the distributive property on the second parentheses. Multiply 7 by X and -6. Your result is 7x - 42. Again, you can eliminate the parentheses. 

Your inequality now reads 1-3x-3+7x-42>0

The next step is to combine your like terms. Like terms are terms that have the same variable and the same degree (exponent). 

The like terms on the left side are the constants: 1, -3, and -42. Let's combine the negative integers first. -42 and -3 make -45. Add 1 and your result is -44

You can rewrite the inequality to read -3x +7x - 44 >0


Now combine the remaining like terms: -3x and 7x. The commutative property allows us to switch the order of these terms without changing the result: 7x -3x. The result is 4x.

Your inequality now reads 4x -44 >0

Almost there! We want to solve for the variable x, so we need to isolate it, or get it by itself on one side of the inequality. To do that, we will use inverse operations to "undo" what is happening to the x. 

X is being multiplied by 4 and X is subtracting 44. 

To "undo" subtracting 44, you should add 44 to both sides of the inequality. 4x -44 + 44 > 0 + 44. The constants eliminate each other on the left, and the right side becomes 44.

4x > 44


X is being multiplied by 4. To "undo" multiplying by 4, you should divide both sides of the inequality by 4.

4x /4 > 44/4 =

x > 11

This is your solution set. To check your work, substitute a value greater than 11 into your inequality. If the inequality is true, values than 11 work. You should also substitute a value less than 11 into your inequality. That should make the inequality false. 

Check:  Let x = 20

1 - 3(12+1) + 7(12-6) > 0 =

1 - 3(13) + 7(6) > 0 = 

1- 39 + 42 > 0 =

-38 + 42 > 0 =

4 > 0 is true, so x can be greater than 11. 

 

Check: Let x = 10

1 -3(10+1) + 7(10-6) >0 =

1- 3(11) + 7(4) > 0 =

1 - 33 + 28 >  0 = 

-32 + 28 > 0 =

-4 > 0 is not true. Therefore, x cannot be less than 11. 

 

I hope this helped!

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