Inequality Notes class 8

 

Inequality Notes class 8

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 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.

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