WAVY CURVE METHOD
Wavy Curve method is probably the most important toolkit to find the domain of an inequality. Not much speaking, Let's get into it :
How to Solve inequality using the Wavy Curve method :
Okay, I will be explaining with the help of an illustrative example. Simply explaining it theoretically would be difficult and hence, explaining with the help of an example would do good.
Consider an inequality (x - 3)(x - 4)²(x-5)³> 0
We are going to solve the above inequality using Wavy Curve Method.
For this,
1. First step is that you must put each factor equal to zero i.e. if the inequality is given of the form
Put,
This will give you many values of x (a1, a2, a3.....upto an) based on each factor. These points a1, a2, a3......an are called the critical points. So, by making each and every factor equal to zero, our aim is to find out the critical points that would make each factor 0.
Applying this rule to above example, We have,
(x - 3) = 0 and (x-4)² = 0 and (x-5)³=0 This would give you three critical points 3, 4 and 5.
2. Plot these critical points on a number line in ascending order. For our above example, we will plot critical points 3, 4 and 5 on the number line
3. We take the rightmost value and put a value greater than this in the entire function. If the sign of function is positive, we start drawing wavy curve from above the number line.. Otherwise, we draw the wavy curve from below the number line. For above example, the righmost value is 5 , we will check the sign of function by putting a value greater than 5. Let's check with 6. Putting this value 6 in our function (x - 3)(x - 4)²(x-5)³, we have +ve sign. So, we start drawing wavy curve from above the number line. The starting of drawing wavy curve would look as in the figure below.The red line indicates the wavy curve.
4. Now comes the important rule. You now have to check on the exponents or powers of each and every factor. If the power of the factor is odd, you must change the side of wavy curve [above / below the number line] otherwise you must continue the wavy curve on the same side of the number line. This sentence would make you confused, so, let's check on this with the example, then only this will be clear to you.
So, here is how it is done -->
STEP 1 : Our function gives +ve sign when we put the value 6 (>5) in our function. So we start drawing curve from above the axis. This I have told you as rule 3 already.
STEP 2 : So, we have reached the number 5. We have got this number from the factor (x-5)³. Now, check the exponent/power of this factor. It is 3. 3 is odd and so, we have to change the side of wavy curve. Initially it was above the number line, so changing its side would shift the wavy curve to below the number line. So,the wavy curve would look something like this :
STEP 3 : Okay, Now, we have reached the number 4. We got this number from the factor (x - 4)². Now, check the exponent/power of this factor. It is 4. 4 is even and so, we have to retain the side of wavy curve i.e. the wavy curve will remain on the same side below the axis. It will not go above the axis. So, the next diagram would look something like this :
STEP 4 : Okay, Now, we have reached the number 3. We got this number from the exponent (x-3). Now, check the exponent/power of this factor. It is 1. 1 is odd and so, we have to change the side of our wavy curve. Initially it is below the number line, now it would go above the number line. The wavy curve would look something like this :
5. So, after you have drawn the wavy curve, you simply need to find the domain using the sign of inequality. If the sign of inequality is > or ≥ , you must choose the values that lie above the number line. And If the sign of inequality is < or ≤, you have to choose the values that lie below the number line.
Wavy Curve method is probably the most important toolkit to find the domain of an inequality. Not much speaking, Let's get into it :
How to Solve inequality using the Wavy Curve method :
Okay, I will be explaining with the help of an illustrative example. Simply explaining it theoretically would be difficult and hence, explaining with the help of an example would do good.
Consider an inequality (x - 3)(x - 4)²(x-5)³> 0
We are going to solve the above inequality using Wavy Curve Method.
For this,
1. First step is that you must put each factor equal to zero i.e. if the inequality is given of the form
Put,
This will give you many values of x (a1, a2, a3.....upto an) based on each factor. These points a1, a2, a3......an are called the critical points. So, by making each and every factor equal to zero, our aim is to find out the critical points that would make each factor 0.
Applying this rule to above example, We have,
(x - 3) = 0 and (x-4)² = 0 and (x-5)³=0 This would give you three critical points 3, 4 and 5.
2. Plot these critical points on a number line in ascending order. For our above example, we will plot critical points 3, 4 and 5 on the number line
3. We take the rightmost value and put a value greater than this in the entire function. If the sign of function is positive, we start drawing wavy curve from above the number line.. Otherwise, we draw the wavy curve from below the number line. For above example, the righmost value is 5 , we will check the sign of function by putting a value greater than 5. Let's check with 6. Putting this value 6 in our function (x - 3)(x - 4)²(x-5)³, we have +ve sign. So, we start drawing wavy curve from above the number line. The starting of drawing wavy curve would look as in the figure below.The red line indicates the wavy curve.
4. Now comes the important rule. You now have to check on the exponents or powers of each and every factor. If the power of the factor is odd, you must change the side of wavy curve [above / below the number line] otherwise you must continue the wavy curve on the same side of the number line. This sentence would make you confused, so, let's check on this with the example, then only this will be clear to you.
So, here is how it is done -->
STEP 1 : Our function gives +ve sign when we put the value 6 (>5) in our function. So we start drawing curve from above the axis. This I have told you as rule 3 already.
STEP 2 : So, we have reached the number 5. We have got this number from the factor (x-5)³. Now, check the exponent/power of this factor. It is 3. 3 is odd and so, we have to change the side of wavy curve. Initially it was above the number line, so changing its side would shift the wavy curve to below the number line. So,the wavy curve would look something like this :
STEP 3 : Okay, Now, we have reached the number 4. We got this number from the factor (x - 4)². Now, check the exponent/power of this factor. It is 4. 4 is even and so, we have to retain the side of wavy curve i.e. the wavy curve will remain on the same side below the axis. It will not go above the axis. So, the next diagram would look something like this :
STEP 4 : Okay, Now, we have reached the number 3. We got this number from the exponent (x-3). Now, check the exponent/power of this factor. It is 1. 1 is odd and so, we have to change the side of our wavy curve. Initially it is below the number line, now it would go above the number line. The wavy curve would look something like this :
5. So, after you have drawn the wavy curve, you simply need to find the domain using the sign of inequality. If the sign of inequality is > or ≥ , you must choose the values that lie above the number line. And If the sign of inequality is < or ≤, you have to choose the values that lie below the number line.
For our above example, the sign of inequality is > and so we choose the values of inequality that lies above the number line. The region enclosed represent the values of our inequality.
For our question, we have sign of inequality as > so we choose values above number line. The red lines represent those values. Look at leftmost, the red curve lies from -∞ to 3. So, we have x ε (-∞ , 3). So also the red curve also lies from 5 to +∞. So, we have x ε (5, +∞). Combining both the results, we have, x ε (-∞ , 3) U (5 , +∞).
