Solving Inequalities
Do you want to know more about inequalities? What are inequalities? How can you solve inequalities?
Inequalities are equations where the values are not equal. It uses the symbols not equal to (≠), greater than (>), greater than or equal to (≥), less than(<) or less than or equal to (≤). The solution to inequality is like solving with an equal sign. Here are the examples of inequalities
x < 4, x + 2 > 5, x2 – x -2> 3 and other.
In graphing inequality, if it was a line, the symbol of inequality used is important. If the equation used greater than (>) or less than (<), the line in the graph is dashed. If the symbol used is greater than or equal to (≥) or less than or equal to (≤) it will be a solid line.
In solving inequalities with one variable or one unknown value, let’s use this example:
4x – 2 > 2 (x + 2) it has one unknown, which is x, and the inequality symbol is greater than (>). Between the two equations, the first one must have greater value to satisfy the equation.
4x – 2 > 2 (x + 2) Treat the inequality like it is equal.
4x – 2 = 2 (x + 2) First, use distribution property to expand the second equation. It will be
2(x) + 2(2) Then solve.
4x – 2 = 2x + 4 Combined similar term.
4x – 2x = 2 + 4
2x = 6 Divide both sides by 2.
x = 3
Since the value of x can be any number higher than 3, this number line represents it. The shaded part is the possible value of x.
Solving Inequalities in two variables
In solving inequalities with two unknown values, the inequality symbol acts like equality.
Let’s have this example: x – y > -2
To solve, let’s assume that the value of x = 0. It can be any number with a low value.
x = 0
x – y = -2
x – y = -2 Substitute x = 0
0 – y = -2
-y = -2 Divide both sides with -1
y = 2
Solve for the value of y if y = 0.
x – y = -2 Substitute y = 0
x – 0 = -2
x = -2
If the value of x is 0, then the ordered pair is (0, 2). If the value of y is 0, then (-2, 0).
Let’s graph the equation.
x – y > -2 Since the symbol of inequality is greater, then the line in the graph must be dashed.
How to know which part to shade? The shaded part must satisfy the equation.
Let’s have a test point.
x = 0 and y = 0
x – y > -2
0 – 0 > -2
0 > -2
Since 0 > -2, the statement is true. Shade the part that satisfies the statement, which contains the point 0.
Solving Quadratic Inequalities
Quadratic inequality is inequality whose function is second degree.There are ways to solve quadratic inequality. These are the factoring, using the quadratic formula, and by completing the square. However, factoring is commonly used.
Lets have this example: x^2 -2x-15 > 0 and used factoring.
x^2 -2x-15 > 0.
x^2 -2x-15 = 0 In factoring, think of a two number in which the product is -15 and if subtracted is -2. The possible pair is 3 and -5.
(x + 3) (x-5) = 0 Get the value of both x.
x + 3 = 0 Subtract both sides by 3.
x = -3
x – 5 = 0 Add 5 on both sides.
x=5
Plot the value of x in a number line to get all the possible values of x, then do the test check. In the test check, choose any number on the number line. Let’s use -5, 0 and 6.
x = -5
x^2 -2x-15 > 0 Substitute the value of x.
(-5)^2 -2(-5) – 15 > 0
25 + 10 – 15 > 0
35-15 > 0
20 > 0
Since the statement is true, all the numbers from -3 are the possible solution.
x = 0
(0)^2 – 2(0) – 15 > 0
-15 > 0
The statement is false, which means all the numbers between -3 and 5 are not the solution.
x = 6
(6)^2 – 2(6) – 15 > 0
36 – 12 – 15 > 0
36 – 27 > 0
9 > 0
9 has a higher value than 0, so the statements is correct. All the numbers from 9 are possible solutions.