There is a certain set of properties that stand true for any symbol of inequality that is used in an expression. Sona of those properties are as follows;
1. Transitivity: this is the relation that is established in the hm three values.
If a ≥ b, and b ≥ c, then a ≥ c
2. Addition or subtraction property: If the same value is added to or subtracted from both sides of an inequality expression, the symbol of inequality will remain the same.
If a < b, then a + x < b + x
3. Multiplication or division property: If a positive quantity is used to multiply or divide the expression on both sides, then the symbol of inequality remains the same. If the value is negative, then the symbol changes.
If a < b, m < 0, then a × m > b × m
If a < b, m > 0, then a × m < b × m
While graphing inequalities, we have to keep the following things in mind.
While writing the solution of an inequality in the interval notation, we have to keep the following things in mind.
A linear inequality is defined as an inequality of the form where the symbol ‘I’ represents any of the inequalities < , >, ≥ , .
Example: Solve the inequality 2(x – 3) –1 > 3(x – 2)– 4(x + 1)
>>> 2x – 7 > 3x – 6 – 4x – 4
>>> 3x > –3.
Hence, x > –1
This can be represented in mathematical terms as (–1,+∞)
A quadratic inequality is defined as an inequality of the form:
ax2 + bx + c I 0 (a 0) where the symbol I represents any of the inequalities <, >, ≥,
For a quadratic expression of the form ax2 + bx + c,
(b2– 4ac) is defined as the discriminant of the expression and is often denoted as D. i.e. D = b2– 4ac
The following cases are possible for the value of the quadratic expression:
Case 1: If D < 0
In other words, we can say that if D is negative then the values of the quadratic expression takes the same sign as the coefficient of x2.
This can also be said as
If D < 0 then all real values of x are solutions of the inequalities ax2 + bx + c > 0 and ax2+ bx + c ≥ 0 for a > 0 and have no solution in case a < 0.
Also, for D < 0, all real values of x are solutions of the inequalities ax2+ bx + c < 0 and ax2 + bx + c 0 if a < 0 and these inequalities will not give any solution for a > 0.
Case 2: D = 0
If the discriminant of a quadratic expression is equal to zero, then the value of the quadratic expression takes the same sign as that of the coefficient of x2 (except when x = –b/2a at which point the value of the quadratic expression becomes 0).
We can also say the following for D = 0:
Case 3: D > 0
If x1 and x2 are the roots of the quadratic expression then it can be said that:
Example: x2 – 5x + 6 > 0
Solution: (a) The discriminant D = 25 – 4 x 6 > 0 and a is positive (+1); the roots of the quadratic expression are real and distinct: x1
= 2 and x2= 3. By the property of quadratic inequalities, we get that the expression is positive outside the interval [2, 3]. Hence, the solution is x < 2 and x > 3.
We can also see it as x2– 5x + 6 = (x – 2) (x – 3) and the given inequality takes the form (x – 2) (x – 3) > 0.
The solutions of the inequality are the numbers x < 2 (when both factors are negative and their product is positive) and also the numbers x > 3 (when both factors are positive and, hence, their product is also positive).
Answer: x < 2 and x > 3.
Example: Solve the system of inequalities:
3x – 4 < 8x + 6
2x – 1 > 5x – 4
11x – 9 15x + 3
Solution: We solve the first inequality:
3x – 4 < 8x + 6
–5x < 10
x > –2
It is fulfilled for x > –2.
Then we solve the second inequality
2x – 1 > 5x – 4
–3x > – 3
x < 1
It is fulfilled for x < 1.
And, finally, we solve the third inequality:
11x – 9 15x + 3
–4x 12
x ≥ – 3
It is fulfilled for x ≥ – 3. All the given inequalities are true for – 2 < x < 1.
| x | a, where a > 0 means the same as the double inequality
This result is used in solving inequalities containing a modulus.
Example: | 2x – 3 | 5
This is equivalent to –5 2x – 3 5
i.e. 2x – 3 ≥ –5 and 2x – 35
2x ≥ –2 x 4
x ≥ –1
The solution is
–1 x 4.
184 videos|131 docs|110 tests
|
1. What is the significance of graphing inequalities in mathematics? |
2. How is writing inequalities in interval notation beneficial in solving mathematical problems? |
3. How can one effectively solve linear inequalities in one unknown? |
4. What is the approach to solving quadratic inequalities in mathematics? |
5. How are inequalities containing a modulus addressed in mathematical problems? |
|
Explore Courses for CAT exam
|