Solving g(x)-h(x) <0: Find a, b, c, d

• MHB
• Manal
In summary, to solve the inequality g(x)-h(x)<0, you need to consider three cases: x<0, 0<x<1, and x>1. For each case, you will get a cubic inequality to solve, and the solution will be in the form of intervals, denoted by a, b, c, and d.
Manal
We have :
f(x) =(1-x) ÷x, x is in IR*
g(x)=|f(x)|
h(x) =4-x^2
Solve g(x)-h(x) <0 (not with a graph)
( I tried solving so , S=]a, b[ U ]c,d[., but i don't know what are a, b, c and d. I would appreciate the help a bunch)

Since $$\displaystyle \left|\frac{1-x}{x}\right|=\frac{|1-x|}{|x|}$$, absolute values can be eliminated on each of the intervals $(-\infty,0)$, $(0,1)$ and $[1,\infty)$ where $1-x$ and $x$ have definite signs. On each interval you get a cubic inequality.

$$\displaystyle f(x)= \frac{1- x}{x}= \frac{1}{x}- 1$$ is 0 at x= 1 and is not continuous at x= 0. Those are the only places f can change sign. If x= -1, f(-1)= -2 so f(x) is negative for all x less than 0. If x= 1/2 f(1/2)= 2- 1= 1 so f(x) is positive for all x between 0 and 1. Finally, if x= 2, f(x)= 1/2- 1= -1/2 so x is negative for all x greater than 1.

g(x)= |f(x)|= 1- 1/x for x< 0
g(x)= 1/x- 1 for 0< x< 1
g(x)= 1- 1/x for 1< x.

If x< 0, g(x)- h(x)= 1- 1/x- 4- x^2= -x^2- 1/x- 3 for x< 0 so you want to solve -x^2- 1/x- 3< 0. Multiplying by -x, which is positive, x^3+ 3x+ 1< 0.

If 0< x< 1, g(x)- h(x)= 1/x- 1- 4- x^2= -x^2+ 1/x- 5 so you want to solve -x^2+ 1/x- 5< 0. Multiplying by -x, which is negative, x^3+ 5x- 1> 0.

If x> 1, g(x)- h(x)= 1-1/x- 4- x^2= -x^2- 1/x- 3 so you want to solve -x^2- 1/x- 3<0. Multiplying by -x, which is negative, x^3+ 3x+ 1> 0.

1. What is the purpose of solving g(x)-h(x) <0 and finding a, b, c, d?

The purpose of solving g(x)-h(x) <0 and finding a, b, c, d is to identify the values of the variables that satisfy the inequality. This can help in determining the range of values for which the given expression is negative, and can be useful in various applications such as optimization problems or finding the roots of a function.

2. How do I solve g(x)-h(x) <0?

To solve g(x)-h(x) <0, you need to first rearrange the expression to isolate the variables on one side and the constants on the other. Then, you can use algebraic methods such as factoring, completing the square, or using the quadratic formula to find the values of the variables that satisfy the inequality.

3. What are the steps to finding a, b, c, d when solving g(x)-h(x) <0?

The steps to finding a, b, c, d when solving g(x)-h(x) <0 depend on the specific expression and the methods used to solve it. However, some common steps include rearranging the expression, factoring, setting each factor equal to zero, and solving for the variables.

4. Can I use a graph to solve g(x)-h(x) <0 and find a, b, c, d?

Yes, you can use a graph to solve g(x)-h(x) <0 and find a, b, c, d. You can plot the given expression on a graph and identify the points where the graph crosses the x-axis (where the expression equals zero). These points can help in determining the values of the variables that satisfy the inequality.

5. Are there any restrictions or limitations when solving g(x)-h(x) <0 and finding a, b, c, d?

There may be some restrictions or limitations when solving g(x)-h(x) <0 and finding a, b, c, d, depending on the specific expression and the methods used to solve it. For example, if the expression involves square roots, there may be restrictions on the values of the variables to avoid taking the square root of a negative number. It is important to carefully consider any potential restrictions or limitations when solving inequalities and finding values for the variables.

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