Interval method is a special algorithm designed to solve complex inequalities of the form f(x) > 0. The algorithm consists of 5 steps:
- Solve the equation f(x) = 0. Thus, instead of an inequality, we get an equation that is much simpler to solve;
- Mark all obtained roots on the coordinate line. Thus, the straight line will be divided into several intervals;
- Find the multiplicity of the roots. If the roots are of even multiplicity, then draw a loop above the root. (A root is considered a multiple if there are an even number of identical solutions)
- Find out the sign (plus or minus) of the function f(x) on the rightmost interval. To do this, it is enough to substitute into f(x) any number that will be to the right of all the marked roots;
- Mark the signs at the remaining intervals, alternating them.
After this, all that remains is to write down the intervals that interest us. They are marked with a “+” sign if the inequality was of the form f(x) > 0, or with a “−” sign if the inequality was of the form f(x)< 0.
In the case of non-strict inequalities (≤ , ≥), it is necessary to include in the intervals points that are a solution to the equation f(x) = 0;
Example 1:
Solve inequality:
(x - 2)(x + 7)< 0
We work using the interval method.
Step 1: replace the inequality with an equation and solve it:
(x - 2)(x + 7) = 0
The product is equal to zero if and only if at least one of the factors equal to zero:
x - 2 = 0 => x = 2
x + 7 = 0 => x = -7
We got two roots.
Step 2: We mark these roots on the coordinate line. We have:
Step 3: we find the sign of the function on the rightmost interval (to the right of the marked point x = 2). To do this, you need to take any number that more number x = 2. For example, let's take x = 3 (but no one forbids taking x = 4, x = 10 and even x = 10,000).
f(x) = (x - 2)(x + 7)
f(3)=(3 - 2)(3 + 7) = 1*10 = 10
We get that f(3) = 10 > 0 (10 is a positive number), so we put a plus sign in the rightmost interval.
Step 4: you need to note the signs on the remaining intervals. We remember that when passing through each root the sign must change. For example, to the right of the root x = 2 there is a plus (we made sure of this in the previous step), so there must be a minus to the left. This minus extends to the entire interval (−7; 2), so there is a minus to the right of the root x = −7. Therefore, to the left of the root x = −7 there is a plus. It remains to mark these signs on the coordinate axis.
Let's return to the original inequality, which had the form:
(x - 2)(x + 7)< 0
So the function must be less than zero. This means that we are interested in the minus sign, which appears only on one interval: (−7; 2). This will be the answer.
Example 2:
Solve inequality:
(9x 2 - 6x + 1)(x - 2) ≥ 0
Solution:
First you need to find the roots of the equation
(9x 2 - 6x + 1)(x - 2) = 0
Let's collapse the first bracket and get:
(3x - 1) 2 (x - 2) = 0
x - 2 = 0; (3x - 1) 2 = 0
Solving these equations we get:
Let's plot the points on the number line:
Because x 2 and x 3 are multiple roots, then there will be one point on the line and above it “ a loop”.
Let's take any number less than the leftmost point and substitute it into the original inequality. Let's take the number -1.
Don’t forget to include the solution to the equation (found X), because our inequality is not strict.
Answer:
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