5. Step 1: Transpose all the variable terms on LHS.
6. Step 2: Transpose all the variable terms on LHS.
7. Step 3: Simplify the terms in LHS.
8. Step 4: Make the coefficient of x positive in the numerator and denominator.
9. Step 5: Equate both the numerator and denominator to zero and find the values of x and critical points.
10. Step 6: Plot these critical points on the number line, which will divide the real line into three regions.
11. Step 7: Mark the sign of the functions over the respective intervals.
12. Step 8: Write solution regions in the form of intervals to get the required solution sets of the given inequality.

6. A linear inequality in two variables represents a half plane geometrically. Types of half planes.

7. In order to identify the half plane represented by an inequality, take any point (a, b) (not on line) and check whether it satisfies the inequality or not. If it satisfies the inequality, then the inequality represents the half plane and subsequently shades the region which contains the point; otherwise, the inequality represents the half plane which does not contain the point within it. For convenience, the point (0, 0) is preferred.

8. If an inequality is of the type ax + b ≥ c or ax + b  c, i.e slack inequality, then the points on the line ax+ b = c are also included in the solution region.

Solution of slack inequality

9. If an inequality is of the form ax + by > c or ax + by < c, then the points on the line ax + by = c are not to be included in the solution region.

Solution of strict inequality

10. To represent x < a (or x > a) on a number line, put a circle on the number a and dark line to the left (or right) of the number a.

11. To represent x ≤ a (or x ≥ a) on a number line, put a dark circle on the number a and dark line to the left (or right) of the number x.

12. Steps to represent the linear inequality in two variables graphically

Step 1: Rewrite the inequality as linear equation, i.e. ax + by = c.

Step 2: Put x = 0 to get y-intercept of the line, i.e. (0, c/b).

Step 3: Put y = 0 to get x-intercept of the line, i.e. (c/a, 0).

Step 4: Join the two points, each on X-axis and Y-axis to get the graphical representation of the line.

Step5: Choose a point (x₁, y₁) in one of the planes, i.e. either to the left or right or upper or lower half of the line, but not on the line.

Step 6: If (x₁, y₁) satisfies the given inequality, then the required region is that particular half plane in which (x₁, y₁) lie.

On the other hand, if (x₁, y₁) does not satisfy the given inequality, then the required solution region is the half plane which does not contain (x₁, y₁).

13. Linear inequalities represent regions; regions common to the given inequalities will be the solution region.

Similar to linear equations, there can be cases of overlapping of regions or no common regions for the given inequalities.

14. To solve a system of inequalities graphically

• Change the sign of equality to inequality and draw the graph of each line.

• Shade the region for each inequality.

• Common region to all the inequalities is the solution.

15. A linear inequality divides the plane into two half planes, while a quadratic inequality is represented by a parabola which divides the plane into different regions.

Region represented by the inequality x² + 5x + 6 > 0

16. Interval Notations

Open Interval: The interval which contains all the elements between a and b excluding a and b. In set notations:

(a, b) = {x : a < x < b}

Closed interval: The interval which contains all the elements between a and b and also the end points a and b is called the closed interval.

[a, b] = {x : a  x  b}

Semi open intervals:

(a, b) = {x : a  x < b} includes all the elements from a to b including a and excluding b.

[a, b] = {x : a < x  b} includes all the elements from a to b excluding a and including b.