Solving the Inequality (x-1) / (x+2) < 1: Step-by-Step Guide

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In summary, the expression "Abs((x-1) / (x+2)) < 1" represents an inequality where the absolute value of the quotient of (x-1) divided by (x+2) is less than 1. The absolute value is important to ensure the result of the quotient is always positive. In a real-life scenario, this expression can represent a ratio between two quantities. Any value of x between -2 and 1, excluding those two values, will satisfy this inequality. This expression cannot be solved in the traditional sense, but can be graphed, tested with different values of x, or simplified algebraically to find the range of values that satisfy it.
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I've tried squaring both sides, then moving the RHS to the LHS, then factorizing according to a2-b2=(a+b)(a-b).

I simplified, and got
((2x+1)/(x+2))((-3/(x+2)) < 0

Then there are 2 cases:
((2x+1)/(x+2)) < 0 and ((-3/(x+2)) > 0

or

((2x+1)/(x+2)) > 0 and ((-3/(x+2)) < 0

I'm not really sure how to go from here...
Please reply ASAP!

 
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  • #2
No need to square.

Don't forget: [itex]|x| < y[/tex], means [itex]\pm x < y[/tex]
 
  • #3
thanks for telling me! now i see how much easier it is :)
 

1. What is the meaning of "Abs((x-1) / (x+2)) < 1"?

The expression "Abs((x-1) / (x+2)) < 1" is an inequality that represents a mathematical statement. It means that the absolute value of the quotient of (x-1) divided by (x+2) is less than 1. In simpler terms, it is saying that the value of (x-1) divided by (x+2) is smaller than 1, regardless of the value of x.

2. What is the significance of the absolute value in this expression?

The absolute value in this expression is used to ensure that the result of the quotient is always positive. This is important because the inequality is comparing the value of the quotient to 1, and without the absolute value, the result of the quotient could be negative, making the comparison inaccurate.

3. How can this expression be interpreted in a real-life scenario?

This expression can be interpreted in a real-life scenario as a representation of a ratio between two quantities. For example, the expression "Abs((x-1) / (x+2)) < 1" can be used to represent the ratio of the number of apples (x) that are ripe (x-1) to the total number of apples (x+2) in a basket. The inequality then means that the ratio of ripe apples to total apples must be less than 1.

4. What values of x satisfy this inequality?

Any value of x that makes the absolute value of the quotient (x-1) / (x+2) less than 1 will satisfy this inequality. This includes all values of x between -2 and 1, excluding those two values. For example, x = 0.5 would satisfy the inequality because the absolute value of the quotient (0.5-1) / (0.5+2) is 0.33, which is less than 1.

5. How can this expression be solved?

This expression cannot be solved in the traditional sense because it is an inequality and not an equation. Instead, it can be graphed on a number line or a coordinate plane to visualize the values of x that satisfy the inequality. Another method is to plug in different values of x and test if they satisfy the inequality. Additionally, algebraic manipulations can be used to simplify the expression and find the range of values that satisfy the inequality.

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