Solving a Linear Inequation Algebraically is explained with different examples in this article. The linear inequality is the mathematical expression in which two sides are not equal to each other. We can represent the linear inequality algebraic expressions with less than, greater than, or with not equal symbols. The symbols that represent the linear inequations are <, >, ≤, ≥. 10th grade math students can find all about the Rules for Solving Linear Inequations Algebraically, Worked Out Examples on Solving Linear Inequalities in this article explained step by step.
Examples:
(i) Solve the inequation 8x + 14 > 46 and the value of the x.
Solution: Given inequation is 8x + 14 > 46.
Subtract the above equation with 14 on both sides.
8x + 14 – 14 > 46 – 14
8x > 32
Divide the above equation with 8 on both sides.
8x/8 > 32/8
x > 4.
Therefore, the value of x is greater than 4.
(ii) Solve the inequation 24 – 10y ≤ 34 and the value of y.
Solution: Given inequation is 24 – 10y ≤ 34.
Subtract the above equation with 24 on both sides.
24 – 24 – 10y ≤ 34 – 24
-10y ≤ 10
Divide the above equation with -10 into both sides. The inequality reverses on multiplying or dividing both sides by -1.
-10y/-10 ≤ 10/-10
y ≥ -1
Therefore, the value of y is equal to -1 or greater than -1.
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Working Rules Based on the Laws of the Inequality
Here is the list of rules associated with solving linear inequalities algebraically. Check out them and learn how to approach solving such kinds of problems on linear inequalities in the below sections.
Rule I: Transferring a Positive Term
In an algebraic linear inequation, if we move the positive term to the other side of the equation, then the term becomes negative. That means the sign of the term becomes negative and we need to subtract it on the other side of the equation.
Examples:
(i) 6x + 7 > 9
Now, move the number 10 to the right side. Then, the linear inequation becomes
6x > 9 – 7
(ii) 14x + 4 ≤ 58
Now, move the number 4 to the right side. Then, the linear inequation becomes
14x ≤ 58 – 4
(iii) 28 ≥ 6x + 22
Now, move the number 22 to the left side. Then, the linear inequation becomes
28 – 22 ≥ 6x
Rule II: Transferring a Negative Term
In an algebraic linear inequation, if we move the negative term to the other side of the equation, then the term becomes positive. That means the sign of the term becomes positive and we need to add it on the other side of the equation.
Examples:
(i) 6x – 10 > 9
Now, move the number 10 to the right side. Then, the linear inequation becomes
6x > 9 + 10
(ii) 14x – 4 ≤ 58
Now, move the number 4 to the right side. Then, the linear inequation becomes
14x ≤ 58 + 4
(iii) 28 ≥ 6x – 22
Now, move the number 4 to the right side. Then, the linear inequation becomes
28 +22 ≥ 6x
Rule III: Multiplication/Division by a Positive Number
Multiply or divide the positive number with each term of the given linear inequation. If we multiply or divide the positive number of the given linear inequation, the sign remains the same.
Examples:
If x is a positive term and a <b, then
a < b —> ax < bx and \(\frac { a }{ x } \) < \(\frac { b }{ x } \),
a > b —> ax > bx and \(\frac { a }{ x } \) > \(\frac { b }{ x } \),
a ≤ b —> ax ≤ bx and \(\frac { a }{ x } \) ≤ \(\frac { b }{ x } \),
a ≥ b —> ax ≥ bx and \(\frac { a }{ x } \) ≥ \(\frac { b }{ x } \),
Take a ≤ 10
Multiply 5 on both sides of the above equation.
5a ≤ 50
Take 2a ≤ 10
Divide 2 on both sides of the above equation.
2a/2 ≤ 10/2
a ≤ 5.
Rule IV: Multiplication/Division by a Negative Number
Multiply or divide the negative number with each term of the given linear inequation. If we multiply or divide the negative number of the given linear inequation, the sign will change.
Examples:
If x is a negative term and a < b, then
a < b —> ax > bx and \(\frac { a }{ x } \) > \(\frac { b }{ x } \),
a > b —> ax < bx and \(\frac { a }{ x } \) < \(\frac { b }{ x } \),
a ≤ b —> ax ≥ bx and \(\frac { a }{ x } \) ≥ \(\frac { b }{ x } \),
a ≥ b —> ax ≤ bx and \(\frac { a }{ x } \) ≤ \(\frac { b }{ x } \),
Take a ≤ 10
Multiply -5 on both sides of the above equation.
-5 × a ≥ -5 × 10
Take 2a ≤ 10
Divide -2 on both sides of the above equation.
-2a/2 ≥ -10/2.
Rule V: Changing the Sign of Each Term
If the sign of each term present on the linear equation changed on both sides, then the sign of inequality will get reversed.
Examples:
(i) – x > 20
Now, change the sign on both sides.
x < -20
(ii) 10t ≤ 38
Now, change the sign on both sides.
-10t ≥ -38
(iii) -18k < – 10
Now, change the sign on both sides.
18k > 10.
Rule VI: Changing the Sign of Each Term
If the terms present on the linear inequations are positive or negative, then by taking their reciprocals, their sign of inequality will reverse.
Examples:
(i) x > y then 1/x > /1/y
(ii) x ≤ y then 1/x ≤ /1/y
(iii) x ≥ y then 1/x ≥ /1/y.
How to Solve a Linear Inequation Algebraically?
Check out the method of Solving a Linear Inequation Algebraically here. Let us take an algebraic expression ax + b > cx + d. Let us see the step-by-step procedure to solve an algebraic expression now.
Step I: In the first step of solving, we need to move all the variables onto one side and move constants on another side. For example, if ax + b > cx + d is our linear inequation, then we need to move x terms on one side and move constants on another side i.e,
ax + b > cx + d
Now, move the x terms to the left side and constant to the right side.
ax – cx > d – b.
Step II: You must turn the given inequation into px > q form.
Step III: Divide or multiply the terms that are attached with the variables to find the values of variables.
Step IV: Finally, you will get the value of the variable.
Algebra: Solving Linear Inequalities Examples
Find the practice questions on solving a linear inequation algebraically and solve all of them on your own. Follow the rules to solve linear inequations algebraically and check the answers after you solve them.
1. Choose the correct answer from the below options. The linear inequation is 2x + 5 > 9.
(i) x > 2
(ii) x < 2
(iii) x ≤ 2
(iv) x ≥ 2.
Solution:
Given inequation is 2x + 5 > 9.
Now, subtract 5 on both sides of the above equation.
2x + 5 – 5 > 9 – 5
2x > 4
Now, divide the above equation with 2 into both sides.
2x/2 > 4/2
x > 2.
Therefore, the correct answer is (i) x > 2.
2. Find the correct option. The linear inequation is 3x – 7 < 11.
(i) 3x – 11 > 7
(ii) 3x > 11 + 7
(iii) 3x < 11 + 7
(iv) 3x – 7 > 11
Solution:
Given inequation is 3x – 7 < 11.
Now, move the 7 to the right side of the above equation. – 7 will turn into 7.
3x – 7 < 11
3x < 11 + 7
Therefore, the correct answer is (iii) 3x < 11 + 7
3. Which two linear equation is similar to the given linear inequation is 3x ≤ 4 from the below options.
(i) 3x ≥ 4
(ii) 6x ≤ 8
(iii) 9x ≤ 8
(iv) 4x ≥ 3.
Solution:
Given inequation is 3x ≤ 4.
Now, multiply the above equation with 2 into both sides.
6x ≤ 8
Therefore, the correct answer is (ii) 6x ≤ 8.
4. Choose the correct answer from the below options and find the value of x. The linear inequation is -3x < 9.
(i) x < 3
(ii) x > -3
(iii) x ≥ 9
(iv) x ≥ 3.
Solution:
Given inequation is -3x < 9.
Now, divide the above equation with -3 into both sides. The inequality reverses on multiplying or dividing both sides by -1.
-3x/-3 < 9/-3
x > -3
Therefore, the correct answer is (ii) x > -3. The value of x is greater than -3.