Inequalities-Multi Step Inequalities

In mathematics, an inequality is used to show a relation between two things that are not in equal quantity. These are separated by greater than (>), less than (<), greater than or equal (≥), and less than or equal to (≤) signs. Inequality can be of different types, such as one-variable inequality, two-variable inequality, compound inequality, and multi-step inequality.

Solving an inequality is similar to solving an equation. Thus, in multi-step inequality, the same number is added, subtracted, multiplied, or divided into both sides. Employing the same number on both sides maintains the balance of inequality. Hence, to solve a multi-step inequality with precision, you must know how to perform different mathematical operations in an inequality.

 Rules for Mathematical Operations in Multi-Step Inequalities

 There are four operations that are carried out on multi-step inequalities. These operations are addition, subtraction, multiplication, and division. The different rules for performing inequality operations are as follows:

The rule for addition in inequalities

According to the addition rule, adding the same number on each side of the inequality doesn’t reverse the inequality. Thus, adding the same number on both sides of the inequality produces equivalent inequality.

Thus, if w > z, then w + a > z + a; and if w < z. then w + a < z + a.

The rule for subtraction in inequalities

According to the rule of subtraction for linear inequalities, subtracting the same number from each side of an inequality produces an equivalent inequality. Therefore, the sign of inequality doesn’t change.

If w < z, then w – a < z – a; and if w > z, then w – a > z – a.

The rules for addition and subtraction of inequalities are easy to remember, and there are fewer chances that a student will make a mistake in the application of rules for addition or subtraction. But the rules for multiplication and division are a bit tricky, and students should be careful while applying them in solving multi-step inequalities.

The rule for multiplication in inequalities

According to the rule for multiplication of linear inequality, multiplication by a positive number on both sides of the inequalities produces an equivalent inequality. That is,

If w > z and a > 0, then w × a > z × a. Similarly, if w < z and a > 0, then w × a < z × a.

However, when an inequality is multiplied by a negative number, then the final inequality is not equivalent inequality. It reverses.

If w > z and a < 0, then w × a < z × a. Similarly, if w < z and a < 0, then w × a > z × a.

The rule for division in inequalities

The rule for division of linear inequalities is similar to the rule for multiplication in linear inequalities. Thus, the division of an inequality with a positive number doesn’t change the sign of inequality. That is,

If w > z and a > 0, then (w / a) > (z / a). Similarly, suppose if w < z and a > 0, then (w/ a) < (z / a)

But when the same inequality is divided by a negative number, then the inequality gets reversed. Thus,  

If w > z and a < 0, then (w / a) < (z / a). Similarly, if w < z and a < 0, then (w / a) > (z / a)

Implementing the Mathematical Operation Rules in Solving Multi-step Inequalities.

 The basic step to solving a multi-step inequality involves isolating the variable. Thus, by performing addition, subtraction, multiplication, or division, the variable should be on one side while the rest must be on the other side. To get a better idea about solving multi-step inequality, an example is given below.

Solve:  9x5 -7 ≥ -3x+12

• The first in solving the above equation is to add both sides by 3x. Adding 3x to RHS will eliminate -3x and the number 12 will only be left. Thus,

:  9x5 -7+3x ≥ -3x+3x+12

:  9x5 -7+3x ≥12

• The second step is to add 7 to both sides of the inequality. Remember, adding a number or variable doesn’t change the sign of inequality.

 9x5 -7+3x+7 ≥12+7

 9x5 +3x ≥19

After eliminating 7 from the inequality, you need to take LCM of 9x5 +3x.

9x5 +3x ≥19

24x5 19

• Now multiply both the sides of inequalities with 5. Multiplying the LHS by 5 will eliminate the 5 from the denominator of 24x5 .

24x5 ≥19

5×24x5 ≥19 ×5

24x95

• Now divide both sides of the inequality by 24.

24x24 ≥ 9524

x ≥3.95 or x ≥4

• To make things easy to interpret, we will take x 4 as the final answer. After getting the final answer, you can plot the result on the number line.

 Conclusion

A common mistake that most students make while solving inequalities is the use of the correct sign. Most students get confused inputting the correct inequality sign when they multiply or divide a multi-step inequality by a negative number. Thus, the final solution they get is completely wrong because changing the inequality sign changes the meaning of the final answer. Thus, while solving inequalities, a person should stay attentive and watchful. Moreover, if the students have a thorough knowledge of the rules of mathematical operations for inequalities, then they completely diminish the chances of making a mistake while solving inequalities.