The replacement set is the set from which the values of the variable (involved in the inequation) are chosen. A number picked from the replacement set that solves the given inequation is called a solution to an inequation. The solution set of an inequation is the set of all solutions to the inequation.

When two numbers are compared on a number line based on their value, numerical inequality occurs. When one expression is more than or less than another, it is called algebraic inequality. This article will learn about Replacement Set and Solution Set theory with examples.

Replacement Set and Solution Set: Overview

The set from which values of the variable involved in the inequality are chosen is called the replacement set. Generally, we use either \(N\) (set of natural numbers; \(1,2,3,4,….)\) or \(W\) (set of whole numbers; \(0,1,2,,3,….)\) or \(I\) (set of integers; \(…-3,-2,-1,0,1,2,3,….)\) or \(R\) (set of real numbers) as the replacement set. Also, the solution to an inequation is a number that, when substituted for the variable, makes the inequation true. The set of all solutions of the given inequation is called the solution set of the inequation. The solution set is the subset of the replacement set.

Example: If \(\left\{ {1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10,\,11} \right\}\) is the replacement set, then find the solution set of the inequality \(x>5.\)

Solution: Given, \(x>5\)

Therefore, the solution set is \(\left\{ {6,\,7,\,8,\,9,\,10,\,11} \right\}.\)

Examples

The set from which the value of the variable \(x\) is chosen is called the replacement set, and its subset, whose elements satisfy the given inequation, is called the solution set.

Example: Let the given inequation be \(x<3,\) if; 

1. The replacement set \(=N,\) the set of natural numbers, the solution set \( = \left\{ {1,\,2} \right\}\)
2. The replacement set \(=W,\) the set of whole numbers, the solution set \[ = \left\{ {0,\,1,\,2} \right\}\]
3. The replacement set \(=Z\) or \(I,\) the set of integers’, the solution set \( = \left\{ {…….,\, – 2,\, – 1,\,0,\,1,\,2} \right\}\) 
   Although, if the replacement set is the set of real numbers, the solution set could only be explained in the set-builder form, i.e. \(\left\{ {x:x \in R\,{\rm{and}}\,x < 3} \right\}.\)

Representation of Solution Set on Number Line

Use the following regulations to represent the solution of a linear inequation in one variable on the number line:

1. If the inequation involves \(\geq\) or \(\leq\), then draw a filled circle or dark circle \((•)\) on the number line to show that the number corresponding to the filled circle or dark circle is included in the solution set.

For example, \(x \leq 2\) and \(x \in R\)

If \(2\) is also included, i.e., \(x≤2\), then the circle will be darkened, and the graph will be as shown below:

2. If the inequation involves \(>\) or \(<,\) then draw an unfilled circle or blank circle on the number line to show, the number corresponding to the unfilled circle or blank circle is excluded from the solution set.

For example, \(x<2\) and \(x∈R\)

The number \(2\) is encircled, and the circle is not darkened to show that \(2\) is not included in the graph.

Example for combining inequations:

Solve and graph the solution set of \(3 x+6 \geq 9\) and \(-5 x>-15\) where \(x \in R\).

Solution: \(3 x+6 \geq 9 \Rightarrow 3 x \geq 9-6\)

\(\Rightarrow 3 x \geq 3 \Rightarrow x \geq 1\)

And, \(-5 x>-15 \Rightarrow \frac{-5 x}{-5}<\frac{-15}{-5} \Rightarrow x<3\)

Graph for \(x \geq 1:\)

Graph for \(x<3:\)

Therefore, graph of the solution set of \(x≥1\) and \(x<3\)

\(=\) Graph of points common to both \(x≥1\) and \(x<3\) is: 

Replacement Set and Solution Set notes

Statements such as \(x<3, x+5 \leq 7,2 x-3>8,3 x+5 \geq 11, \frac{x-3}{2}<2 x+1\) are called linear inequations in one variable. In general, a linear inequation in one variable can always be written as \(a x+b<0, a x+b \leq 0, a x+b>0\) or \(a x+b \geq 0\) where \(a\) and \(b\) are real numbers, \(a \neq 0\)

Replacement set: The set from which values of the variable (involved in the inequation) are chosen is called the replacement set.

Solution set: A solution to an inequation is a number (chosen from the replacement set) that makes the inequation true when substituted for the variable. The set of all solutions of an inequation is called the solution set of the inequation.

For example, consider the inequation \(x<4.\)

Replacement set	Solution set
\(\left\{ {1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10} \right\}\)	\(\left\{ {1,\,2,\,3} \right\}\)
\(\left\{ { – 1,\,0,\,1,\,2,\,5,\,8} \right\}\)	\(\left\{ { – 1,\,0,\,1,\,2} \right\}\)
\(\left\{ { – 5,\,10} \right\}\)	\(\left\{ { – 5} \right\}\)
\(\left\{ {5,\,6,\,7,\,8,\,9,\,10} \right\}\)	\(\Phi\)

Note that the solution set depends upon the replacement set.

Solved Examples

Q.1. If the replacement set is the set of integers (\(I\) or \(Z\)), between \(-6\) and \(8,\) find the solution set of \(15-3x>x-3.\)
Ans: \(15-3 x>x-3\)
\(\Rightarrow-3 x-x>-3-15\)
\(\Rightarrow-4 x>-18\)
\(\Rightarrow \frac{-4 x}{-4}<\frac{-18}{-4}\) (Division by a negative number reverse the sign of inequality),
\(\Rightarrow x<4.5\)
Since the replacement set is the set of integers between \(-6\) and \(8.\)
Therefore, solution set \( = \left\{ { – 5, – 4, – 3, – 2, – 1,0,1,2,3,4} \right\}\)

Q.2. If the replacement set is the set of real numbers \(\left( R \right),\) find the solution set of \(5-3x<11.\)
Ans: \(5-3 x<11\)
\(\Rightarrow-3 x<11-5\)
\(\Rightarrow-3 x<6\)
\(\Rightarrow \frac{-3 x}{-3}>\frac{6}{-3}\)  (Division by a negative number reverse the sign of inequality),
\(\Rightarrow x>-2\)
Since the replacement set is the set of real numbers \(R.\)
Therefore, solution set \( = \left\{ {x:x > – 2\,{\rm{and}}\,x \in R} \right\}\)

Q.3. List the solution set of \(50 – 3\left( {2x – 5} \right) < 25,\) given that \(x \in W.\) Also, represent the solution set obtained on a number line.
Ans: \(50-3(2 x-5)<25\)
\(\Rightarrow 50-6 x+15<25\)
\(\Rightarrow-6 x<25-65\)
\(\Rightarrow-6 x<-40\)
\(\Rightarrow \frac{-6 x}{-6}>\frac{-40}{-6}\)(Division by a negative number reverse the sign of inequality),
\(\Rightarrow x>6 \frac{4}{6}\)
Therefore, the required solution set \( = \left\{ {7,\,8,\,9,\,…..} \right\}\)
And the required number line is: 

Q.4. Solve the inequation \(37 – \left( {3x + 5} \right) \ge 9x – 8\left( {x – 3} \right)\) represent the solution set on the number line. \(x\) is a whole number.
Ans: Given, \(37-(3 x+5) \geq 9 x-8(x-3)\)
\(\Rightarrow 37-3 x-5 \geq 9 x-8 x+24\)
\(\Rightarrow 32-3 x \geq x+24\)
\(\Rightarrow 32-3 x-32 \geq x+24-32\) (Subtracting \(32\) from both sides),
\(\Rightarrow-3 x \geq x-8\)
\(\Rightarrow-3 x-x \geq x-8-x\) (Subtracting \(x\) from both sides),
\(\Longrightarrow-4 x \geq-8\)
\(\Rightarrow \frac{-4 x}{-4} \leq \frac{-8}{-4}\) (dividing both sides by \(-4)\),
\(\Rightarrow x \leq 2\)
Therefore, solution set \(=(-\infty, 2)\)
On the number line, it can be represented as: 

Q.5. Write the solution of the following inequation in the set notation form \(5x-10≤2x+2.\)
Ans: \(5 x-10 \leq 2 x+2\)
\(\Rightarrow 5 x-10+10 \leq 2 x+2+10\) (adding \(10\) on both sides),
\(\Rightarrow 5 x \leq 2 x+12\)
\(\Rightarrow 5 x-2 x \leq 2 x+12-2 x\) (subtracting \(2x\) from both sides),
\(\Rightarrow 3 x \leq 12\)
\(\Rightarrow \frac{3 x}{3} \leq \frac{12}{3}\) (dividing both sides by \(3)\),
\(\Rightarrow x \leq 4\)
Hence, the solution set is \(\left\{ {x:x \le 4,\,x \in R} \right\}\) or \((-\infty, 4)\)

Summary

In this article, we learned about the explanation of replacement set and solution set, the example of replacement set and solution set, representation of solution set on the number line, replacement set and solution set notes, solved examples on replacement set and solution set, FAQs on replacement set and solution set.

This article’s learning outcome is that we understood how to find the solution set from the given replacement set and how to represent the solution set on the number line.

FAQs

Q.1. What is the replacement set and solution set?
Ans: Replacement set: The replacement set is the set from which the values of the variable (involved in the inequation) are chosen.
Solution set: A number (from the replacement set) that makes the inequation true when substituted for the variable is called a solution to an inequation. The solution set of an inequation is the collection of all solutions to the inequation.

Q.2. How do you write a solution set?
Ans: The solution set for an equation is the collection of all the solutions to that equation. If there are no solutions to an equation, we write \(\Phi\) for the solution set, also called the “null set” (or empty set).
Example: If the replacement set is the set of integers lying between \(-3\) and \(10,\) find the solution set of \(14-5 x \geq 3 x-40\)
Solution: Given, \(14-5 x \geq 3 x-40\)
\(\Rightarrow 14-5 x+5 x \geq 3 x-40+5 x(\) Add \(5 x)\)
\(\Longrightarrow 14 \geq 8 x-40\)
\(\Rightarrow 14+40 \geq 8 x-40+40\) ( Add \(40)\)
\(\Longrightarrow 54 \geq 8 x\)
\(\Rightarrow 8 x \leq 54 \quad(\because a \geq b \Rightarrow b \leq a)\)
\(\Rightarrow \frac{8 x}{8} \leq \frac{54}{8}\) (Divide by \(8\))
\(\Longrightarrow x \leq 6.75\)
But the replacement set is the set of integers lying between \(-3\) and \(10.\)
i.e……\(\left\{ { – 2,\, – 2,\,0,\,1,\,2,\,….,\,9} \right\}\)
Therefore, the solution set is \(\left\{ { – 2,\, – 2,\,0,\,1,\,2,\,3,\,4,\,5,\,6} \right\}.\)

Q.3. What is the replacement set in linear inequalities?
Ans: Replacement set: The set from which values of the variable (involved in the inequation) are chosen is called the replacement set.
For example, consider the inequation \(x<3.\)
Replacement set \( = \left\{ {1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10} \right\}\)
Solution set \( = \left\{ {1,\,2} \right\}.\)

Q.4. What is the solution set of the inequality?
Ans: Solution set: A solution to an inequation is a number (chosen from the replacement set) that makes the inequation true when substituted for the variable. The set of all solutions of an inequation is called the solution set of the inequation.
For example, consider the inequation \(x>5.\)
Replacement set \( = \left\{ {1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10} \right\}\)
Solution set \( = \left\{ {6,\,7,\,8,\,9,\,10} \right\}.\)

Q.5. What is an example of linear inequality?
Ans: Linear Inequalities: In mathematics, an inequality occurs when two mathematical expressions or two integers are compared in a non-equal way. 
Statements such as \(y<2, x+3 \leq 2, y-2>6,3 x+5 \geq 11, \frac{z-5}{3}<z+3\) are few examples of linear inequations in one variable. In general, a linear inequation in one variable can always be written as \(a x+b<0, a x+b \leq 0, a x+b>0\) or \(a x+b \geq 0\) where \(a\) and \(b\) are real numbers, \(a \neq 0 .\)

Q.6. Are inequalities and inequations the same?
Ans: When used as nouns, inequality denotes an unjust, unequal situation, whereas inequation denotes a claim that two statements are not equivalent.

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