Inequality Notes class 8

Introduction:

        यह अध्याय कक्षा 8 गणित का एक महत्वपूर्ण भाग है, जिसमें Inequalities (असमानताएँ) के बारे में बताया गया है। इस अध्याय में हम सीखते हैं कि दो राशियों के बीच “बराबर” (=) के अलावा “कम” (<) या “अधिक” (>) का संबंध कैसे दर्शाया जाता है। इसमें विभिन्न प्रकार की असमानताओं को समझकर उन्हें हल करने की विधियाँ सरल और step-by-step तरीके से प्रस्तुत की गई हैं, ताकि विद्यार्थी इस विषय को आसानी से समझ सकें।

What is Inequality?

Inequality refers to a mathematical statement that shows the relationship between two expressions that are not equal. In inequalities, the two expressions are connected using inequality symbols.

Inequality Symbols:

• > (Greater Than): Shows that the value on the left is larger than the value on the right.
  Example: $5 > 3$
  

• < (Less Than): Shows that the value on the left is smaller than the value on the right.
  Example: $3 < 5$
  

• ≥ (Greater Than or Equal To): Indicates that the value on the left is either greater than or equal to the value on the right.
  Example: $5 \geq 3$
  

• ≤ (Less Than or Equal To): Indicates that the value on the left is either less than or equal to the value on the right.
  Example: $3 \leq 5$
  

• ≠ (Not Equal To): Indicates that two values are not equal.
  Example: $5 \neq 3$
  

Solving Inequalities:

Just like solving equations, inequalities can be solved to find the value(s) of the variable. However, there are a few key points to remember when solving inequalities:

• Step 1: Simplify the inequality, if needed, by removing parentheses or combining like terms.

• Step 2: Isolate the variable (just like in equations) on one side of the inequality.

• Step 3: Perform operations on both sides (addition, subtraction, multiplication, division) but remember the rule for multiplication and division by negative numbers.

Important Notes

• Inequality में “>”, “<”, “≥”, “≤” जैसे चिन्हों का उपयोग किया जाता है।
• यदि दोनों पक्षों में एक ही संख्या जोड़ी या घटाई जाए, तो असमानता का चिन्ह नहीं बदलता।
• यदि दोनों पक्षों को किसी धनात्मक संख्या से गुणा या भाग किया जाए, तो चिन्ह वही रहता है।
• यदि दोनों पक्षों को किसी ऋणात्मक संख्या से गुणा या भाग किया जाए, तो असमानता का चिन्ह उल्टा हो जाता है।
• Inequalities को संख्या रेखा (number line) पर भी दर्शाया जा सकता है।

Important Rule:

• When multiplying or dividing by a negative number, reverse the inequality symbol.
  Example: If $-2x > 4$, then divide by $-2$, and the inequality becomes $x < -2$
  

Example 1:

Solve $3x - 5 \geq 10$

$Solution:$

• Step 1: Add 5 to both sides:
  $3x \geq 15$
  

• Step 2: Divide both sides by 3:
  $x \geq 5$
  

Example 2:

Solve $-2x + 6 < 14$

$Solution:$

• Step 1: Subtract 6 from both sides:
  $-2x < 8$
  

• Step 2: Divide both sides by -2 (flip the inequality):
  $x > -4$
  

Graphing Inequalities

Inequalities can be represented graphically on a number line.

• Open Circle (for strict inequalities): Used when the number is not included in the solution.
  Example: $x < 5$ or $x > 5$ is graphed with an open circle on 5.

• Closed Circle (for non-strict inequalities): Used when the number is included in the solution.
  Example: $x \leq 5$ or $x \geq 5$ is graphed with a closed circle on 5.

Example:

For $x \geq 3$, you draw a closed circle at 3 and shade the number line to the right, indicating all values greater than or equal to 3.

5. Compound Inequalities

These are two or more inequalities combined together to form a range of possible solutions. They are often written in two forms:

• "And" (Intersection): Both conditions must be true at the same time.
  Example: $2 < x \leq 5$ means $x$ is greater than 2 and less than or equal to 5.

• "Or" (Union): At least one of the conditions must be true.
  Example: $x \leq 3 \text{ or } x > 7$ means $x$ is less than or equal to 3, or greater than 7.

Example of "And":

Solve $-3 \leq x < 4$:

• The solution is all numbers from $-3$ to 4, including $-3$, but not including 4.

Example of "Or":

Solve $x < -2 \text{ or } x \geq 5$:

• The solution includes all numbers less than $-2$, or greater than or equal to $5$.

6. Word Problems Involving Inequalities

• Inequalities are also used to solve real-life problems. For example:

Example 1:
A student needs at least 60% in their final exam to pass. If the student has already scored 45% in the midterm, how much do they need to score in the final to pass?
Solution:

Let $x$ be the percentage the student needs to score in the final exam. The inequality is:

$45+x\ge 60$

Solving this:

$x\ge 60-45$

$x \geq 15$

So, the student needs to score at least 15% in the final exam.

Example 2:
A company can produce no more than 500 units of a product in a day. If the company has already produced 375 units, how many more units can they produce?
Solution:

Let $x$ be the number of units they can still produce. The inequality is:

$375+x\le 500$

Solving this:

$x\le 500-375\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$

$x \leq 125$

So, the company can produce no more than 125 units.

1) Solving Simple Linear Inequalities

Example 1:

Solve the inequality:

$$3x-5\ge 7$$

Solution:
Step 1: Add 5 to both sides:

$$3x\ge 12$$

Step 2: Divide both sides by 3:

$$x\ge 4$$

So, the solution is:

$$x\ge 4$$

Example 2:

Solve the inequality:

$$4x+3<19$$

Solution:
Step 1: Subtract 3 from both sides:

$$4x<16$$

Step 2: Divide both sides by 4:

$$x<4$$

So, the solution is:

$$x<4$$

2) Solving and Graphing Linear Inequalities

Example 3:

Solve the inequality:

$$-2x+6>10$$

Solution:
Step 1: Subtract 6 from both sides:

$$-2x>4$$

Step 2: Divide both sides by -2 (remember to flip the inequality sign when dividing by a negative number):

$$x<-2$$

So, the solution is:

$$x<-2$$

Now, you can graph this on a number line, with an open circle at -2 (because the inequality is strict, i.e., "less than") and a shaded region to the left of -2.

3) Solving Compound Inequalities

Example 4:

Solve the inequality:

$$3<2x+4\le 10$$

Solution:
Step 1: Solve the left part of the inequality.

$$3<2x+4\text{Subtract 4 from both sides:}-1<2x$$

Now, divide by 2:

$$-\frac{1}{2}<x$$

Step 2: Solve the right part of the inequality.

$$2x+4\le 10\text{Subtract 4 from both sides:}2x\le 6$$

Now, divide by 2:

$$x\le 3$$

So, the solution is:

$$-\frac{1}{2} < x \leq 3$$

4) Word Problem Involving Inequalities

Example 5:

A concert hall has 400 seats. The number of tickets sold, $t$, cannot exceed 350. Write an inequality to represent this situation and solve for $t$.

Solution:
The number of tickets sold cannot exceed 350, so we write the inequality:

$$t\le 350$$

This means that the maximum number of tickets that can be sold is 350.

5) Rational Inequalities

Example 6:

Solve the inequality:

$$\frac{3x}{4}\le 6$$

Solution:
Step 1: Multiply both sides by 4 to eliminate the fraction:

$$3x\le 24$$

Step 2: Divide both sides by 3:

$$x\le 8$$

So, the solution is:

$$x\le 8$$

Practice Questions

Try solving these on your own:

1. Solve: $5x - 7 > 8$
   

2. Solve: $4(x - 1) \geq 12$
   

3. Solve the compound inequality: $-3 \leq 2x + 1 < 5$
   

4. A movie theater has 150 seats. They want to sell no more than 100 tickets. Write an inequality for the number of tickets sold, $t$.

5. Solve: $\frac{2x}{5} \geq 4$
   

Conclusion

इस अध्याय में हमने Inequalities (असमानताओं) के मूल सिद्धांतों और उन्हें हल करने की विधियों को समझा। यह विषय गणित में तार्किक सोच को विकसित करने में मदद करता है। नियमित अभ्यास से विद्यार्थी इस अध्याय को आसानी से समझ सकते हैं और परीक्षा में अच्छे अंक प्राप्त कर सकते हैं।

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