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The standard form of a quadratic inequality in one variable is f(x) = ax^2 + bx + c > 0 (or < 0). If the inequality is written in another form, transform it into the standard form, with a trinomial in x on the left side and 0 on the right side.
Solving the inequality for x means finding the values of x that make the inequality true. These values of x constitute the solution set of the inequality. The solution set is expressed in terms of intervals. By convention, we use these following symbols, x1 and x2 being the 2 real roots of the equation f(x) = 0:
- (x1 , x2) for the open interval between x1 and x2. The end points x1 and x2 are not included in the solution set.
- [x1 , x2] for the closed interval between x1 and x2. The end points are included in the solution set.
Note. When the inequality has an additional equal sign (greater, less, or equal to), the 2 end points are automatically included in the solution set. The solution set may be a closed interval [x1 , x2]. Or, the solution set may be half closed intervals: (-infinity, x1] and [x2 , + infinity).
There are 3 common methods to solve a quadratic inequality:
1. Using the number-line
2. Using the algebraic method.
3. By graphing.
Approaches in solving quadratic inequalities.
There are 2 approaches in solving quadratic inequalities depending on the chosen method. The test point approach is convenient for the number-line or the graphing method. The second approach uses a theorem on the sign of a trinomial f(x), if the algebraic method is chosen.
Examples of solving quadratic inequalities.
Example 1. Solve: f(x) = 12x^2 + 5x - 72 < 0.
Solution. First, solve the equation f(x) = 0. I advise you to use the Diagonal Sum Method to solve it. Roots have opposite signs. Write down the all-options-line. After elimination of no-fitted options, the remainder probable roots-set is (-8/3 & 9/4) that give as 2 real roots: -8/3 and 9/4. Next, solve the inequality f(x) < 0.
Plot the two real roots -8/3 and 9/4 on the number line. Use the origin as test point. Substitute x = 0 into the inequality. You get: -72 < 0. It is true, then the origin is located on the true segment. The solution set is the open interval: (-8/3 , 9/4). The 2 end points are not included in the solution set.
Example 2. Solve f(x) = 45x^2 + 74x - 55 > 0.
Solution. Use the Diagonal Sum Method to solve f(x) = 0. It gives 2 real roots:
-11/5 and 5/9. Plot these 2 values on the number-line. Substitute x = 0 into the inequality f(x) > 0. You get: -55 > 0. It is not true!. Then, the origin is on the "wrong" segment. The solution set are the two rays. The solution set are: (-infinity , -11/5) or (5/9 , +infinity). The 2 end points are not included.
What else can I help you with?
What is a double inequality?
A double inequality is an inequality where there are two signs, as opposed to one.Ex: an inequality could be 3x < 15A double inequality could be 3x < 15 < x + 20If you'd want to solve that double inequality, you split it into to expressions:3x < 15 and x + 20 > 15Then just solve.x < 5 and x > -5-5 < x < 5
What are some signs of inequality?
Some signs of inequality may be physical or emotional differences.
What does Inequality mean in math?
In mathematics, an inequality is a statement about the relative size or order of two values.
When a quantity is added to or subtracted from both sides of an inequality the resulting inequality will have the same direction as the original one?
When a quantity is subtracted or added from both sides of an inequality, the true difference in value is varied thereby changing the direction of the inequality, but when rather than subtracted or added it is multiplied or divided, it preserves the true difference in value thereby facing the same direction as the initial inequality.
What mean inequality?
Not the same as, or different from.
How are quadratic inequalities different from linear inequalities?
A linear inequality is all of one side of a plane. A quadratic inequality is either the inside of a parabola or the outside.
What is quadratic inequality in one variables?
A quadratic inequality in x is in the standard form of ax^2+bx+c(>or<)d. Ex. 3x^2+5x+1>4
Is yx3cubed plus x2squared a quadratic inequality?
No. This is not an inequality, because you need something > something_else, or less than or 'not equal' or 'greater than or equal', etc. Since it has an x cubed term, it is not a quadratic.
How do you tell a linear inequality from a quadratic inequality?
In a linear inequality the variable is only present raised to the first power (which is usually not explicitly shown). In a quadratic the square of the variable is present (or implied). The square can be implied in an inequality such as x + 1/x < 6 (x not 0) This is equivalent to x2 - 6x + 1 < 0
What is the degree of a quadratic inequality?
The degree of a quadratic inequality is 2. This is because it involves a quadratic expression, typically in the form (ax^2 + bx + c ), where (a), (b), and (c) are constants and (a \neq 0). The inequality can be expressed as (ax^2 + bx + c < 0), (ax^2 + bx + c > 0), or similar forms, all of which are characterized by the highest exponent of the variable being 2.
How do you solve a quadratic inequality when the expression is unfactorable?
Use the quadratic formula for the equality. Then, depending on the coefficient of x2 and the nature of the inequality [>, ≥, ≤, <], determine whether you need the open or closed intervals between the roots or beyond the roots.
How can the change in x affect the change in y in a inequality?
The answer depends on the nature of the inequality: whether it is linear, quadratic or has some other functional form.
How do you describe the graph of solution set of a quadratic inequality?
The graph of the solution set of a quadratic inequality typically represents a region in the coordinate plane, where the boundary is formed by the parabola defined by the corresponding quadratic equation. Depending on the inequality (e.g., (y < ax^2 + bx + c) or (y > ax^2 + bx + c)), the solution set will include points either above or below the parabola. The parabola itself may be included in the solution set if the inequality is non-strict (e.g., ( \leq ) or ( \geq )). The regions of the graph where the inequality holds true are shaded or highlighted to indicate the solution set.
Can a quadratic inequality have a solution set that is all real numbers?
Yes. Consider x2 ≥ 0
What number is a solution of the inequality?
To determine a solution to an inequality, you need to specify the inequality itself. Solutions vary depending on the inequality's form, such as linear (e.g., (x > 3)) or quadratic (e.g., (x^2 < 4)). Once the inequality is provided, you can identify specific numbers that satisfy it. Please provide the inequality for a precise solution.
Is the solution to a quadratic inequality in one variable always a compound inequality?
Yes - except in extreme cases. It can be the whole of the Real Numbers: eg x2 > -3 It can be a single point eg x2 ≤ 0 gives x = 0
What is an inequality containing the word or?
It could be the solution to some quadratic inequalities: for example x2 + 2x - 3 > 0 whose solution is x < -3 or x > 1.
