In this article we will explore how to solve Inequalities with Multiplication and Division in detail and solve some examples related to it. Let's start our learning on the topic "Solving Inequalities with Multiplication and Division."
Table of Content
What are Inequalities?
The equations with inequality symbol are referred to as inequalities. The inequalities compare the quantities that are not equal with the help of the inequality symbols. Some examples of inequalities are x< 9, 2> 1 etc.
Different Inequalities Symbols
The below table represents different inequality symbols.
Inequality Name | Inequality Symbol |
|---|---|
> | |
< | |
≥ | |
≤ |
Solving Inequalities with Multiplication and Division
To solve the inequalities, we multiply or divide same number on both the sides of the inequality and simplify it. We further multiply or divide if needed to solve the inequality.
Solving Inequalities with Multiplication
The solving inequality with multiplication involves checking the sign of the number that is multiplied. If the number multiplied is positive, inequality does not change but if the number is negative inequality changes. To solve inequality with multiplication we multiply same number on both sides of inequality.
Multiplication with Positive Numbers
If we multiply same positive number on both sides of inequality then, the inequality remains same i.e., unchanged. The solving inequality with multiplication of positive number c is represented as:
- a > b then ac > bc
- a < b then ac < bc
- a ≥ b then ac ≥ bc
- a ≤ b then ac ≤ bc
Multiplication with Negative Numbers
If we multiply same negative number on both sides of inequality then, the inequality is reversed or flipped. The solving inequality with multiplication of negative number c is represented as:
- a > b then ac < bc
- a < b then ac > bc
- a ≥ b then ac ≤ bc
- a ≤ b then ac ≥ bc
Solving Inequalities with Division
The solving inequality with division involves checking the sign of the number that is multiplied. If the number divided is positive, inequality does not change but if the number is negative inequality changes. To solve inequality with division we multiply same number on both sides of inequality.
Division with Positive Numbers
If we divide same positive number on both sides of inequality then, the inequality remains same i.e., unchanged. The solving inequality with division of positive number c is represented as:
- a > b then a/c > b/c
- a < b then a/c < b/c
- a ≥ b then a/c ≥ b/c
- a ≤ b then a/c ≤ b/c
Division with Negative Numbers
If we divide same negative number on both sides of inequality then, the inequality is reversed or flipped. The solving inequality with division of negative number c is represented as:
- a > b then a/c < b/c
- a < b then a/c > b/c
- a ≥ b then a/c ≤ b/c
- a ≤ b then a/c ≥ b/c
Read More,
- Solve Inequalities with Addition and Subtraction
- Linear Inequalities
- Quadratic Inequalities
- Compound Inequalities
Solved Examples on Solving Inequalities with Multiplication and Division
Example 1: Solve: (x / 3) < 10
Solution:
(x/3) < 10
Multiply by 3 (3 is positive number so inequality does not flip)
3 × (x/3) < 10 × 3
⇒ x < 30
Example 2: Solve [y /(-4)] > 12
Solution:
[y /(-4)] > 12
Multiply by (-4) [-4 is negative number so inequality is flipped]
[y /(-4)] × (-4) < 12 × (-4)
⇒ y < -48
Example 3: Evaluate (z / 5) > 23
Solution:
(z / 5) > 23
Multiply by 5 (5 is positive number so inequality does not flip)
(z / 5) × 5 > 23 × 5
⇒ z > 115
Example 4: Solve (p/ (-6)) < 2
Solution:
(p/ (-6)) < 2
Multiply by (-6) [-6 is negative number so inequality is flipped]
(p/ (-6)) × (-6) > 2 × (-6)
⇒ p > -12
Example 5: Solve x / 3 ≤ -15
Solution:
x / 3 ≤ -15
Multiply by 3
x ≤ -15 × 3
⇒ x ≤ -45
Example 6: Solve (-3x) ≤ -15
Solution:
(-3x) ≤ -15
Divide by -3
(-3x) / (-3) ≥ -15 / (-3)
⇒ x ≥ 5
Example 7: Solve 8x > 16
Solution:
8x > 16
Divide by 8
x > 16 / 8
⇒ x > 2
Example 8: Solve -12y < 36
Solution:
-12y < 36
Divide by -12
y > 36 / (-12)
⇒ y > -3
Practice Problems on Solving Inequalities with Multiplication and Division
Q1. Solve: (x / 5) < 21
Q2. Solve [y /(-6)] > 15
Q3. Solve (z / 3) > 5
Q4. Solve (p/ (-11)) < 7
Q5. Solve (q / 5) ≥ 4
Q6. Solve (-2x) ≥ -5
Q7. Solve 4x < 20
Q8. Solve (-3x) > 24
Q9. Solve 16x > 48
Q 10. Solve 15x ≤ 30