Solve linear inequalities :
A statement involving variable or variables and the sign of inequality like <, >, ≤ or ≥ is called an inequality.
Here we are going to see how to solve linear inequalities.
Linear equations in one variable :
Let a be a non zero real numbers and x be a variable. Then the inequality of the form ax + b < 0, ax + b ≤ 0, ax + b > 0 and ax + b ≥ 0 are known as linear equations in one variable.
Rules followed in solving linear equations in one variable :
Rule 1 :
Same number may be added to (or subtracted from) both sides of an inequality without changing the sign of inequality.
Rule 2 :
Both sides of an inequality can be multiplied (or divided) both by the same positive real number without changing the sign of inequality. However, the sign of inequality is revered when both sides of an inequality are multiplied or divided by the negative number.
Rule 3 :
Any term of an inequality may be taken to the other side with its sign changed without affecting sings of inequality.
Let us see some examples based on the above concept.
Example 1 :
Solve the following linear inequality
2x - 4 ≤ 0
2x - 4 ≤ 0
Add 4 on both sides
2x - 4 + 4 ≤ 0 + 4
2x ≤ 4
Divide by 2 on both sides
2x/2 ≤ 4/2
x ≤ 2
By graphing this in the number line, we can split this as two intervals. (-∞, 2] and [2, ∞)
Now we need to select one of the values from the above intervals and apply those values instead of x in the given question 2x - 4 ≤ 0.
The values from which interval makes the given inequality true is the solution set.
Let x = 0
2(0) - 4 ≤ 0
- 4 ≤ 0 True
Let x = 3
2(3) - 4 ≤ 0
6 - 4 ≤ 0
2 ≤ 0 False
Hence (-∞, 2] is the solution set.
Linear equations in two variables :
If a, b, c are real numbers, then the equation ax + by + x = 0 is called a linear equation in two variables.Whereas the inequalities ax + by < c, ax + by ≤ c, ax + by > c and ax + by ≥ c are known as linear equations in two variables.
Steps followed in solving linear equations in two variables :
Step 1 :
Convert the given inequality, say ax + by ≤ c, into the equation ax + by = c which represents a straight line in xy plane.
Step 2 :
Put y = 0 in order to find the x-intercept and similarly put x = 0 to find y-intercept. Now draw the straight by using these intercepts.
Step 3 :
Choose a point and substitute its coordinates in the inequality, if the inequality is satisfied, then shade the portion of the plane contains the chosen point, otherwise shade the portion which does not contain the desired solution set.
Example 1 :
Solve the following inequality graphically
2x - y ≥ 1
Consider the linear inequality as linear equation
2x - y = 1
put y = 0
2x - 0 = 1
2x = 1
x = 1/2 = 0.5
put x = 0
2(0) - y = 1
-y = 1
y = -1
By using the above two points we can draw a straight line.Now we need to select a point say (2, -2)
2x - y ≥ 1
2(2) - (-2) ≥ 1
4 + 2 ≥ 1
6 ≥ 1
Since the chosen point satisfies the given inequality, we can shade that portion.
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