Solving Inequality Problems

The phenomena of unequal and/or unjust allocation of resources and opportunities among individuals of a given society is referred to as inequality. To various people and in different settings, the phrase inequality can mean different things. Furthermore, inequality has unique but overlapping economic, social, and geographic dimensions. The disconnect between the moral ethics of equity and social justice on the one hand, and the normative concept of “deservingness” on the other, complicates discussions concerning inequality. In recent years, there has been a growing awareness of inequities within social groupings as well as those that exist across social groups. As a result of this understanding, more people are realizing that inequality is structural and embedded in numerous socioeconomic and political organizations. Highlighting the continuation of unequal and differentiated rights, investigating global and extreme inequality, and evaluating the manifestations and causes of urban disparities are some of geographers’ contributions to the theme of inequality in this regard. 

Definition

The topic of mathematical inequalities is intimately related to optimization techniques. Inequalities are common in mathematics Olympiads, despite the fact that they are often left out of the regular educational route.

Inequalities with Fractions

Equations are frequently discussed in algebra. Equations represent the relationship between two algebraic expressions that is equal. This is a crucial relationship that is frequently used in mathematics. Inequalities, on the other hand, are significant relationships in which one of two expressions (or numbers) is claimed to be more or less than the other.

The inequality ab, for example, says that the value of an is smaller than that of b. As long as the first variable is less than the second, the variables can represent any number or value.

Inequalities with fractions simply mean that the expressions on one or both sides of the inequality have inequalities.   An unbalanced seesaw is a fantastic way to visualize a fractional inequality.

Remember to imagine equations and inequalities as a seesaw. 

Two-Step Inequalities

Two-step inequalities and two-step equations are the same thing. A two-step inequality is one that requires two steps to solve (usually an add/subtract step and a multiply/divide step). Solving two-step inequalities including fractions is the same as solving two-step inequalities without fractions.

Steps to Solving Inequalities with Fractions

As previously stated, the steps for solving inequalities with fractions are the same as for solving inequalities of any kind. In general, there will be two steps required:

• To isolate the variable term, first add or subtract a constant from both sides of the inequality

• Second, to totally isolate the variable, multiply or divide both sides of the inequality by the variable’s coefficient

• The final concern is the only distinction between solving equations and inequalities. The inequality sign must be switched around if a negative amount was multiplied/divided on both sides of the inequality

Rules for Inequalities

If your finite math teacher asks you to answer a linear inequality, you can apply the same rules you’d use to solve linear equations. However, there are two major exceptions, which you’ll learn about below.

The following list contains all of the rules you’ll need to know while working with inequalities. Although only the symbol is displayed in this list, the same criteria apply to any inequality, including >, ≤, and ≥.

• If a< b, then a + c <b + c. The orientation of an inequality sign is not changed by adding the same integer to each side

• If a< b, then a – c < b – c. The orientation of an inequality sign is not changed by subtracting the same integer from each side

• If a < b and c are both positive numbers, then a· c< b· c. The orientation of an inequality sign is not changed by multiplying either side by a positive amount

• If both a<b and c are positive numbers, then

                 a/c < b/c

Dividing each side of an inequality by a positive number has no effect on the inequality symbol’s direction.

• If a< b is negative and c is positive, then a. c > b .c. When each side of an inequality is multiplied by a negative value, the inequality sign is reversed

• If both a and b are negative numbers, then

         a/c > b/c

The direction of an inequality symbol is reversed when each side is divided by a negative number.

Solving Inequalities step by step

The steps for resolving inequalities are as follows:

Step 1: Make an equation out of the inequality.

Step 2: Determine one or more values for the equation.

Step 3: On the number line, represent all of the values.

Step 4: Use open circles to indicate all omitted values on the number line.

Step 5: Determine the intervals.

Step 6: Pick a random number from each interval, substitute it into the inequality, and see if the inequality is satisfied.

Step 7: The solutions are the satisfied intervals.

However, we normally use algebraic operations like addition, subtraction, multiplication, and division to solve simple inequalities (linear). Consider the following illustration:

2x + 3 > 3x + 4

3x and 3 are subtracted from both sides.

2x – 3x > 4 – 3

-x > 1

By adding -1 to both sides,

X < -1

The “>” symbol has been replaced with the “<” symbol. Why? This is because both sides of the inequality have been multiplied by a negative integer. A basic linear inequality can be solved using the procedure described above. However, we must use the following procedure to solve any other complex inequality.

Conclusion

Inequalities are used in mathematics to compare the relative sizes of values. They’re useful for comparing numbers, variables, and other algebraic expressions. Consider the following scenarios: highway speed restrictions, minimum credit card payments, the quantity of text messages you can send each month from your cell phone, and the time it will take to commute from home to school. Mathematical inequalities can be used to represent all of these.