Multi-step inequalities are similar to multi-step equations, but instead of finding the value of the variable that makes the equation true, you are finding a range of values that makes the inequality true. These types of inequalities involve more than one step to solve, typically requiring operations like combining like terms, distributing, and isolating the variable on one side of the inequality sign.

In this article, we will learn what Multi-Step Inequalities are, and understand how to solve their problems.

What are Multi-Step Inequalities?

Multi-step inequalities are mathematical expressions involving inequalities (such as <, ≤, >, ≥) that require more than one step to solve. These inequalities involve combining like terms, using the distributive property, or performing operations on both sides of the inequality to isolate the variable.

For example: 2(3x-4) ≥ 5x+2, 3.5 > x, etc are known as inequalities problems.

Properties to Multi-Step Inequalities

Addition and Subtraction Properties:

• You can add or subtract the same number from both sides of the inequality without changing the direction of the inequality.
• If a<b, then a+c<b+c and a-c<b-c.

Multiplication and Division Properties:

• You can multiply or divide both sides of the inequality by a positive number without changing the direction of the inequality.
• If a<b and c>0, then ac<bc and \[Tex]\frac{a}{c} < \frac{b}{c}[/Tex]​.
• If you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality.
• If a<b and c<0, then ac>bc and [Tex]\frac{a}{c} > \frac{b}{c}[/Tex]​.

Distributive Property:

• Apply the distributive property to eliminate parentheses: a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac.
• This helps in simplifying the inequality to a more manageable form.

Combining Like Terms:

• Combine like terms on each side of the inequality to simplify the expression.
• This involves adding or subtracting coefficients of the same variable and constants.

Isolating the Variable:

• Aim to isolate the variable on one side of the inequality by using addition, subtraction, multiplication, or division.
• This often requires performing multiple steps, such as moving terms involving the variable to one side and constants to the other side.

Example: Solve the inequality [Tex]3(x – 2) + 5 \leq 2(x + 1) + 4[/Tex].

Solution:

• Distribute: [Tex]3x – 6 + 5 \leq 2x + 2 + 4[/Tex]
• Combine like terms: [Tex]3x – 1 \leq 2x + 6[/Tex]
• Isolate the variable: [Tex]3x – 2x – 1 \leq 6[/Tex]
• Add 1 to both sides: [Tex]x \leq 7[/Tex]

Solution is [Tex]x \leq 7[/Tex].

Ssteps in Solving Multi-Step Inequalities

Various steps involved in solving multi step inequalities are:

Step 1: Simplify Both Sides: Combine like terms on both sides of the inequality if necessary.

Step 2: Use the Distributive Property: If there are parentheses, apply the distributive property to eliminate them.

Step 3: Isolate the Variable: Move the variable terms to one side of the inequality and the constant terms to the other side. This often involves adding, subtracting, multiplying, or dividing both sides of the inequality.

Step 4: Solve the Inequality: Perform the necessary operations to isolate the variable on one side of the inequality.

Step 5: Flip the Inequality Sign: If you multiply or divide both sides of the inequality by a negative number, remember to reverse the inequality sign.

Examples on Multi-Step Inequalities

Example 1: Solve 3x-4≤2(x+5)-x.

Solution:

Distribute the 2 on the right side:[Tex]3x – 4 \leq 2x + 10 – x[/Tex]

Combine like terms on the right side:[Tex]3x – 4 \leq 2x – x + 10[/Tex]

[Tex]3x – 4 \leq x + 10[/Tex]

Subtract x from both sides to isolate the terms with x on one side:[Tex]3x – x – 4 \leq x – x + 10[/Tex]

[Tex]2x – 4 \leq 10[/Tex]

Add 4 to both sides to isolate the term with x:[Tex]2x – 4 + 4 \leq 10 + 4[/Tex]

[Tex]2x \leq 14[/Tex]

Divide both sides by 2 to solve for x:[Tex]\frac{2x}{2} \leq \frac{14}{2}[/Tex]

[Tex]x \leq 7[/Tex]

Solution to the inequality is: [Tex]x \leq 7[/Tex].

Example 2: Solve 3(2x-4) ≤ 5x+6.

Solution:

Distribute the 3: 6x-12≤5x+6.

Move the 5x to the left side by subtracting 5x from both sides:

6x – 5x – 12 ≤ 6

Simplify:

x – 12 ≤ 6

Add 12 to both sides:

x ≤ 18

So, the solution is x ≤ 18

Example 3: Solve 2x+3 ≤ 4x-5

Solution:

Subtract 2x from both sides: 3≤2x-5

Add 5 to both sides: 8≤2×8

Divide by 2: 4≤x4

x ≥ 4x

Example 4: Solve 5-3x > 7+x

Solution:

Subtract x from both sides: 5-4x>7

Subtract 5 from both sides: -4x>2

Divide by -4 and flip the inequality: x<-1/2

Solution: x<-1/2

Example 5: Solve 4(x-2) ≥ 2(x+3).

Solution:

Distribute: 4x-8≥2x+6

Subtract 2x2x2x from both sides: 2x-8≥6

Add 8 to both sides: 2x≥14

Divide by 2: x≥7

x ≥ 7

Example 6: Solve 6x+8 < 2(3x+5)

Solution:

Distribute: 6x + 8 < 6

Subtract 6x from both sides: 8<10

Inequality holds for all x.

All x are solutions.

Example 7: Solve 3(x+4)-2 ≤ 5(x-2)

Solution:

Distribute: 3x+12-2 ≤ 5x-10

Combine like terms: 3x+10 ≤ 5x-10

Subtract 3x3x3x from both sides: 10 ≤ 2x-10

Add 10 to both sides: 20 ≤ 2x

Divide by 2: 10 ≤ x

x ≥ 10x

Example 8: Solve 7 – 2(x-3) > 4x-8

Solution:

Distribute: 7-2x+6>4x-8

Combine like terms: 13-2x>4x-8

Add 2x to both sides: 13>6x-8

Add 8 to both sides: 21>6x

Divide by 6: 21/6>x

Simplify: 3.5>x

x < 3.5x

Read More:

• Trigonometric Identities
• Algebraic Identities
• Compound Inequalities
• Graphical Solution of Linear Inequalities in Two Variables

Practice Questions on Multi-Step Inequalities

Q1. Solve 2(3x-2) – 4 ≤ 5x + 6

Q2. Solve 5(2x-1) > 4(x+3)

Q3. Solve 2x+3(4-x) ≤ 3(x-2)+7

Q4. Solve 6-(x+2) ≥ 3x+1

Q5. Solve 4(x+3) – 5 ≤ 6x-2

Q6. Solve 8x-3(2x+4) > 7

Q7. Solve 3(x+5)-2x ≥ 4x-7

Q8. Solve 5-4(x+2) ≤ 3(x-1)

Q9. Solve 7x-2(3x+1) > 4

Q10. Solve 2(x-3)+5 ≤ 4(x+1)-2

FAQs on Multi-Step Inequalities

What is a multi-step inequality?

A multi-step inequality is an inequality that requires more than one step to solve. It involves simplifying expressions, combining like terms, and performing operations on both sides of the inequality to isolate the variable.

When do you reverse the inequality sign?

You reverse the inequality sign when you multiply or divide both sides of the inequality by a negative number. This rule is crucial because it maintains the correct relationship between the sides of the inequality.

What is the difference between solving inequalities and equations?

The main difference is the treatment of the inequality sign. While solving inequalities, you need to remember to reverse the inequality sign when multiplying or dividing by a negative number.

Can inequalities have no solution or all real numbers as a solution?

Yes. An inequality can have no solution if the inequality statement is never true. Conversely, it can have all real numbers as a solution if the inequality statement is always true.

How are absolute value inequalities solved?

To solve absolute value inequalities:

• For ∣x∣<a (where a is positive), rewrite as -a < x < a.
• For x∣>a (where a is positive), rewrite as x < -a or x > a.