Inequalities

Equations in mathematics are not usually about balancing sides with an ‘equal sign.’ Sometimes, it is about a ‘not equal to’ relationship when something is larger than or less than the other. In mathematics, an inequality is a relationship that compares two numbers or other mathematical expressions in a non-equal way. Inequalities are mathematical expressions that fall within the category of algebra. We have to reduce and solve linear inequalities like algebraic equations. Linear is defined as anything resembling a straight line and having only one direction. Non-linear is defined as things that cannot be linear and are not one-directional. The output is not directly proportional to its input. 

Linear Equalities

A linear inequality in two variables is created by substituting the equality sign in a two-variable linear equation with one of the four inequality signs: >,=,>=. Because there are infinitely many possible linear equations, there are infinitely many inequalities. The graphed region containing points that meet the inequality is always half of the plane bounded by the matching line. The line is included when the inequality is = or >=. The line is not included if that is not the case.

• > greater than
• < less than
• ≤ less than or equal to
• ≥ greater than or equal to
• ≠ not equal to
• = equal to

A linear inequality in a linear function that contains an inequality (axe + by c, etc.). This is what you must remember from algebra.

In Rn, linear inequalities are expressions of the form f(x) b, where f is a linear form, x = (x1, x2, x3, x4,…., xn), and b is a constant real integer. These functions can also be represented as g(x) 0, where g represents an affine function.

The matrix inequality Ax=b can be expressed as a mxn matrix, a n x 1 column vector of variables, and a column vector of constants, where A is a mxn matrix, x is a n x 1 column vector of variables, and b is a column vector of constants.

An example of a linear inequality is 

2x+3y≤102x+3y≤10

As long as the expressions on both sides are linear and the relation is not exact equality, it is a linear inequality. Or perhaps you meant it literally:

5x+4y≠205x+4y≠20

As you can see, most values of x and y will satisfy such an inequality. In fact, only values of the form y=5–(5/4)xy=5–(5/4)x do not.

To solve inequality, just solve the corresponding equality by replacing the ≠≠ with == and solve it. That tells you which values do not satisfy the inequality; all other values or combinations of values do.

Solving Inequalities

The basics of solving inequalities are adding (or subtracting) the same amount or multiplying (or dividing) by the same amount on each side of the inequality.

Step 1: Remove fractions by multiplying all terms by the fractions’ least common denominator.

Step 2: Combine like terms on each side of the inequality to simplify.

Step 3: To get the unknown on one side and the integers on the other, add or subtract quantities.

Solving Inequalities in One Step

Take the inequality 2x 6 as an example. To solve this, only one step is required: divide both sides by two. Then we get x 3 as a result. As a result, the inequality’s solution is x 3 (or) (-, 3).

Solving Inequalities in Two Steps

Consider the inequalities -2x + 3 > 6 and -2x + 3 > 6. There are two methods to resolving this. The first step is to deduct 3 from both sides of the equation, yielding -2x > 3. After that, we need to divide both sides by -2, which gives us x -3/2. (Note that we have changed the sign of the inequality). As a result, the inequality’s solution is x -3/2 (or) (-, -3/2).

Reducing Inequalities

Working with reducing inequalities is quite similar to dealing with equations. We normally need to know the domain of the variable when dealing with reducing inequalities. The solution set of reducing inequalities must be calculated in this manner. Let’s imagine our variable is x, which can be found in either an open or closed interval.

1. 10< x > 14 implies open interval and the solution set x=(11,12,13)
2. 10≤ x ≥ 14 implies closed interval and x={10,11,12,13,14}
3. 10< x ≥ 14 implies half open interval and x=(11,12,13,14}

Conclusion

A linear inequality is a mathematical inequality that incorporates a linear function. One of the symbols of inequality is found in linear inequality. In graph form, it depicts data that is not equal.

Always keep in mind, we never get a closed interval in the solution if the symbol is strictly less than or strictly greater than. Because they are not integers to include, we always receive open intervals at or – symbols. When calculating rational inequalities, always write open intervals at excluded values. Only in the case of linear inequality should excluded values be considered.