Inequalities-Compound Inequality

A compound inequality is a statement that is joined by using the word “or” or by the word “and”, and both words have different meanings. The use of the word “and” indicates that both statements are true at the same time

Thus, the solution of the equation containing the word “and” represents the intersection of solutions. Whereas, if the two equations are separated by using the word “or” then it signifies that until one of the statements is true, the complete sentence is not true. 

In compound inequality, the use of the word “and” represents conjunction, while the use of the word “or” represents dis-conjunction. Thus, to make things easy to learn, we can remember the rule by thinking about the word “con” as “with another,” and the word “dis” means “one or another.”

Basic properties of inequalities

There are some basic properties of inequality. These properties can also be used to solve the compound inequality problems. The basic properties for inequalities are as follows:-

1. If a > b, b > c then a > c

2. If a>b the a + m >b + m

3. If a > b then am > bm for m > 0 and for am < bm, m < 0

4. If a > b > 0 then 1a< 1b 

5. If a1 > b1, a2 > b2, …….. an > bn then a1 + a2 + ….. an > b1 + b2 + …… bn for all positive a’s and b’s.

6. If a > b > 0 and n > 0 then an > bn and a(1/n) > b(1/n) 

7. If a > b > 0 and x > 0 then ax > bx

8. If x > y > 0 and a > 1 then ax > ay 

9. If x > y > 0 and 0 < a < 1 then ax < ay

10. If x > y > 0 and a > 1 then logax > logay

11. If x > y > 0 and 0 < a < 1 then logax < logay

Procedure to Solve Compound Inequality

Solving a compound inequality is not tricky if you have basic knowledge about performing mathematical operations in an inequality. An example question is solved below to enhance understanding of the topic.

Example: – Solve for x: 3x+2<14 and 2x-5> -11

• For solving a compound inequality question, you need to solve both the inequalities separately. Since the two inequality mentioned in the question are separated by ‘and’ hence the solution of the system of linear inequalities will be an intersection or overlap of the solution of each inequality.

3x + 2 < 14

Or, 3x < 12

Or, x < 4

2x – 5 > -11

Or, 2x > -6

Or, x > -3

• The solution of the first equation indicates that all the values of x are less than -3. While the solution of the second equation shows that the value of x is greater than -3. 

• Thus, the solution of the system of equations lies between the x > -3 and x < 4. Another way to write the above solution can be to { -3 < x < 4}

Representing compound inequality on a graph

• Since the above equations have ‘and’ in the question, a solution of the two equations needs to be marked between the intersection of the two solutions. To draw the solution on the graph, you need to follow some simple steps.

• Draw an infinite line with the help of a scale on graph paper and mark numbers on it.

• Remember, the numbers to the left side of zero will be negative, while the numbers to the right side of zero will be positive. 

• Now to mark x< 4 on the number line, draw a hollow circle on the number 4 on the number line. 

• After drawing the hollow circle on 4, shade the area which is greater than 4 towards infinity.

• Similarly, mark the -3 on the graph with a hollow circle. After marking -3 on the graph, shade the area which is greater than -3.

• Now you will see that shading will intersect at a common area. This common area is the solution to the compound inequalities.

Rules for drawing an inequality graph

The rules to draw an inequality graph for a one-variable linear equation are pretty simple. However, none of the rules should be missed while drawing the graph. Because with the slightest of carelessness, the final answer can get wrong. The rules for drawing the graph are as follows:-

• Always draw an infinite line for representing one variable linear equation.

• If the equation has a less than ( < ) or greater than ( > ) sign, then a hollow circle should be used to mark the solution on the number line.

For example, the graphical interpretation of -3<x<4 is

• If the equation has a greater than or equal to ( ≥ ) or less than or equal to (≤) sign, then a solid circle is used to represent the solution of the inequality.

For example, the graphical interpretation of –9 ≤ x ≤ 1 is

• Remember, the numbers to the left of zero are negative, while the numbers to the right of zero are positive. Also, there is no strict rule to mark zero in the middle of the number line.

• Draw the number to an accurate scale.

Conclusion

Inequality is a challenging topic for most people because most inequality questions are represented in a word problem. Thus knowing the meaning of the terms “at least”, “at most”, etc., is vital.