Solve Inequalities with Multiplication and Division

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.”

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.

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

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

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

What is an inequality?

An inequality is a mathematical statement that compares two expressions using inequality symbols like <, >, ≤, and ≥.

What are the common inequality symbols?

The common inequality symbols are: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

How do you solve a linear inequality?

To solve a linear inequality, isolate the variable on one side by performing the same operations on both sides.

What happens when you multiply or divide by a negative number?

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.

Can inequalities have multiple solutions?

Yes, inequalities often have multiple solutions, which can be represented as a range or an interval on a number line.

What is a compound inequality?

A compound inequality involves two inequalities connected by “and” or “or”. For “and,” solutions must satisfy both inequalities; for “or,” solutions must satisfy at least one.