Maharashtra Board Class 11 Maths Part 2 Chapter 8 Linear Inequations 8.2 Solutions

Question 1. Solve the following inequations graphically in a a two-dimensional plane
(i) x ≤ -4
Answer: Given, inequation is x ≤ -4
∴ corresponding equation is x = -4
It is a line parallel to Y-axis passing through the point A(-4, 0)
Origin test:
Substituting x = 0 in inequation, we get
0 ≤ -4 which is false.
∴ Points on the origin side of the line do not satisfy the inequation.
So the points on the non-origin side of the line and points on the line satisfy the inequation
∴ all the points on the line and left of it satisfy the given inequation.
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(x \le -4\) का ग्राफिक समाधान दर्शाता है। इसमें Y-अक्ष के समानांतर \(x = -4\) पर एक ऊर्ध्वाधर रेखा खींची गई है, और समाधान क्षेत्र को इस रेखा के बाईं ओर छायांकित किया गया है।
In simple words: To solve \(x \le -4\) graphically, draw a vertical line at \(x = -4\). Since \(0 \le -4\) is false, the solution is the region to the left of this line, including the line itself.

🎯 Exam Tip: When testing the origin for inequalities, if the origin test is false, shade the region opposite to the origin. If it's true, shade the region containing the origin.

(ii) y ≥ 3
Answer: Solution:
Given, inequation is y ≥ 3
∴ corresponding equation is y = 3
It is a line parallel to X-axis passing through point A(0, 3)
Origin test:
Substituting y = 0 in inequation, we get
0 ≥ 3 which is false.
∴ Points on the origin side of the line do not satisfy the inequation
∴ Points on the non-origin side of the line satisfy the inequation.
∴ all the points on the line and above it satisfy the given inequation.
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(y \ge 3\) का ग्राफिक समाधान दिखाता है। इसमें X-अक्ष के समानांतर \(y = 3\) पर एक क्षैतिज रेखा खींची गई है, और समाधान क्षेत्र को इस रेखा के ऊपर छायांकित किया गया है।
In simple words: To solve \(y \ge 3\) graphically, draw a horizontal line at \(y = 3\). Since \(0 \ge 3\) is false, the solution is the region above this line, including the line itself.

🎯 Exam Tip: Remember that lines parallel to the Y-axis are of the form \(x=k\), and lines parallel to the X-axis are of the form \(y=k\).

(iii) y ≤ -2x
Answer: Solution:
Given, inequation is y ≤ -2x
∴ corresponding equation is y = -2x
It is a line passing through origin O(0, 0).
To draw the line, we need one more point.
To find another point on the line, we can take any value of x,
say, x = 2.
∴ substituting x = 2 in y = -2x, we get
y = -2(2)
∴ y = -4
∴ another point on the line is A(2, -4)
Now, the origin test is not possible as the origin lies on the line y = -2x
So, choose a point which does not lie on the line say, (2, 1)
∴ substituting x = 2, y = 1 in inequation, we get
1 ≤ -2(2)
∴ 1 ≤ -4 which is false.
∴ the points on the side of the line y = -2x, where (2, 1) lies do not satisfy the inequation.
∴ all the points on the line y = -2x and on the opposite side of the line where (2, 1) lies, satisfy the inequation
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(y \le -2x\) का ग्राफिक समाधान प्रस्तुत करता है। इसमें मूल बिंदु से गुजरने वाली रेखा \(y = -2x\) खींची गई है, और समाधान क्षेत्र को उस तरफ छायांकित किया गया है जो बिंदु (2,1) के विपरीत है।
In simple words: For \(y \le -2x\), draw the line \(y = -2x\) through the origin. Since the origin is on the line, test a point not on the line, like (2,1). As \(1 \le -4\) is false, shade the region opposite to (2,1).

🎯 Exam Tip: When the origin lies on the boundary line, choose any other convenient point not on the line to test the inequality. Ensure your chosen test point clearly determines the correct shaded region.

(iv) y – 5x ≥ 0
Answer: Solution:
Given, inequation is y – 5x ≥ 0
∴ corresponding equation is y – 5x = 0
It is a line passing through the point O(0, 0)
To draw the line, we need one more point.
To find another point on the line,
we can take any value of x, say, x = 1.
Substituting x = 1 in y – 5x = 0, we get
y - 5(1) = 0
∴ y = 5
∴ Another point on the line is A(1, 5)
Now origin test is not possible as the origin lies on the line y = 5x
∴ choose a point that does not lie on the line, say (3, 2).
∴ substituting x = 3, y = 2 in inequation, we get
2 – 5(3) ≥ 0
∴ 2-10 ≥ 0
∴ -8 ≥ 0 which is false.
∴ the points on the side of line y = 5x where (3, 1) lies do not satisfy the inequation.
∴ the points on the line y = 5x and on the opposite of the line where (3, 2) lies, satisfy the inequation.
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(y - 5x \ge 0\) का ग्राफिक समाधान दर्शाता है। इसमें मूल बिंदु से गुजरने वाली रेखा \(y = 5x\) खींची गई है, और समाधान क्षेत्र को उस तरफ छायांकित किया गया है जो बिंदु (3,2) के विपरीत है।
In simple words: For \(y - 5x \ge 0\), draw the line \(y = 5x\) through the origin. Since the origin is on the line, test a point like (3,2). As \(-8 \ge 0\) is false, shade the region opposite to (3,2).

🎯 Exam Tip: For inequalities passing through the origin, selecting a test point carefully is crucial. Any point not on the line will work to determine the correct half-plane.

(v) x - y ≥ 0
Answer: Solution:
Given, inequation is x - y ≥ 0
∴ Corresponding equation is x - y = 0
It is a line passing through origin O(0, 0)
To draw the line we need one more point.
To find another point on the line, we can take any value of x,
Say, x = 2.
∴ substituting x = 2 in x - y = 0, we get
2- y = 0
∴ y = 2
∴ another point on the line is A(2, 2)
Now origin test is not possible as the origin lies on the line y = x
∴ choose a point which not lie on the line say (3, 1)
∴ substituting x = 3, y = 1 in inequation, we get
3-1≥ 0
∴ 2 ≥ 0 which is true.
∴ all the points on line x – y = 0 and the points on the side where (3, 1) lies satisfy the inequation
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(x - y \ge 0\) का ग्राफिक समाधान प्रस्तुत करता है। इसमें मूल बिंदु से गुजरने वाली रेखा \(x - y = 0\) खींची गई है, और समाधान क्षेत्र को उस तरफ छायांकित किया गया है जहां बिंदु (3,1) स्थित है।
In simple words: For \(x - y \ge 0\), draw the line \(x = y\) through the origin. Since the origin is on the line, test a point like (3,1). As \(2 \ge 0\) is true, shade the region containing (3,1).

🎯 Exam Tip: Lines of the form \(x - y = 0\) or \(x = y\) always pass through the origin. Ensure you choose a test point that clearly distinguishes between the two half-planes.

(vi) 2x - y ≤ -2
Answer: Solution:
Given, inequation is 2x – y ≤ -2
∴ corresponding equation is 2x – y = -2
∴ \( \frac{2x}{-2} - \frac{y}{-2} = \frac{-2}{-2} \)
\( \implies \frac{x}{-1} + \frac{y}{2} = 1 \)
∴ intersection of line with X-axis is A(-1, 0),
intersection of line with Y-axis is B(0, 2)
Origin test:
Substituting x = 0, y = 0 in the given inequation, we get
2(0) – (0) ≤ -2
∴ 0 ≤ -2
which is false.
∴ Points on the origin side of the line do not satisfy the inequation.
∴ Points on the non-origin side of the line satisfy the inequation
∴ all the points on the line and above it satisfy the given inequation.
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(2x - y \le -2\) का ग्राफिक समाधान दिखाता है। इसमें X-अक्ष को A(-1, 0) पर और Y-अक्ष को B(0, 2) पर काटने वाली रेखा \(2x - y = -2\) खींची गई है, और समाधान क्षेत्र को मूल बिंदु के विपरीत दिशा में छायांकित किया गया है।
In simple words: For \(2x - y \le -2\), draw the line \(2x - y = -2\), which has x-intercept (-1,0) and y-intercept (0,2). Since the origin test (\(0 \le -2\)) is false, shade the region opposite to the origin.

🎯 Exam Tip: Converting the linear equation to intercept form (\(x/a + y/b = 1\)) helps in quickly finding the intercepts and drawing the line accurately.

(vii) 4x + 5y ≤ 40
Answer: Solution:
Given, inequation is 4x + 5y ≤ 40
∴ Corresponding equation is 4x + 5y = 40
∴ \( \frac{4x}{40} + \frac{5y}{40} = \frac{40}{40} \)
\( \implies \frac{x}{10} + \frac{y}{8} = 1 \)
∴ Intersection of line with X-axis is A(10, 0)
Intersection of line with Y-axis is B(0, 8)
Origin test:
Substituting x = 0, y = 0 in the inequation, we get
4(0) + 5(0) ≤ 40
∴ 0 ≤ 40 which is true.
∴ all the points on the origin side of the line and points on the line satisfy the given inequation.
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(4x + 5y \le 40\) का ग्राफिक समाधान दर्शाता है। इसमें X-अक्ष को A(10, 0) पर और Y-अक्ष को B(0, 8) पर काटने वाली रेखा \(4x + 5y = 40\) खींची गई है, और समाधान क्षेत्र को मूल बिंदु की ओर छायांकित किया गया है।
In simple words: For \(4x + 5y \le 40\), draw the line \(4x + 5y = 40\), which has x-intercept (10,0) and y-intercept (0,8). Since the origin test (\(0 \le 40\)) is true, shade the region containing the origin.

🎯 Exam Tip: Always clearly label the intercepts on the axes when drawing the line, as it helps in validating the graph. The intercept form is very useful for this.

(viii) \( \left(\frac{1}{4}\right) x + \left(\frac{1}{2}\right) y \le 1 \)
Answer: Solution:
Given, inequation is \( \left(\frac{1}{4}\right) x + \left(\frac{1}{2}\right) y \le 1 \)
∴ corresponding equation is \( \frac{x}{4} + \frac{y}{2} = 1 \)
∴ intersection of line with X-axis is A(4, 0),
intersection of line with Y-axis is B(0, 2)
Origin test:
Substituting x = 0, y = 0 in the given inequation, we get
\( \frac{1}{4}(0) + \frac{1}{2}(0) \le 1 \)
∴ 0 ≤ 1 which is true.
∴ all the points on the origin side of the line and points on the line satisfy the given inequation.
The shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \( \frac{x}{4} + \frac{y}{2} \le 1 \) का ग्राफिक समाधान दिखाता है। इसमें X-अक्ष को A(4, 0) पर और Y-अक्ष को B(0, 2) पर काटने वाली रेखा \( \frac{x}{4} + \frac{y}{2} = 1 \) खींची गई है, और समाधान क्षेत्र को मूल बिंदु की ओर छायांकित किया गया है।
In simple words: For \( \frac{x}{4} + \frac{y}{2} \le 1 \), draw the line with x-intercept (4,0) and y-intercept (0,2). Since the origin test (\(0 \le 1\)) is true, shade the region containing the origin.

🎯 Exam Tip: When an inequation is already in intercept form, identifying the intercepts is straightforward, making line plotting faster. Always verify the inequality direction with a test point.

Question 2. Mr. Rajesh has Rs. 1,800 to spend on fruits for the meeting. Grapes cost Rs. 150 per kg. and peaches cost Rs. 200 per kg. Formulate and solve it graphically.
Answer: Solution:
Let x and y be the number of kgs. of grapes and peaches bought.
The cost of grapes is Rs. 150/- per kg, cost of peaches is Rs. 200/- per kg.
∴ cost of v kg of grapes is Rs. 150x
and the cost of y kg of peaches is Rs. 200y.
Mr. Rajesh has Rs. 1800 to spend on fruits.
∴ the total cost of grapes and peaches must be less than or equal to Rs. 1800.
∴ required inequation is 150x + 200y ≤ 1800
i.e., 3x + 4y ≤ 36 ......(i)
Since the number of kg of grapes and peaches can not be negative
∴ x ≥ 0, y ≥ 0
Now, corresponding equation is 3x + 4y = 36
∴ \( \frac{3x}{36} + \frac{4y}{36} = \frac{36}{36} \)
\( \implies \frac{x}{12} + \frac{y}{9} = 1 \)
∴ the intersection of the line with the X-axis is A(12, 0)
the intersection of the line with the Y-axis is B(0, 9)
Origin test:
Substituting x = 0, y = 0 in inequation, we get
3(0) + 4(0) ≤ 36
∴ 0 ≤ 36 which is true.
∴ all the points on the origin side of the line and points on the line satisfy the inequation.
Also, x ≥ 0, y ≥ 0
∴ the solution set is the points on the sides of the triangle OAB and in the interior of ΔΟΑΒ.
∴ the shaded portion represents the solution set.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(3x + 4y \le 36\) का ग्राफिक समाधान दर्शाता है, साथ ही \(x \ge 0\) और \(y \ge 0\) की शर्तों को भी दिखाता है। इसमें X-अक्ष को A(12, 0) पर और Y-अक्ष को B(0, 9) पर काटने वाली रेखा \(3x + 4y = 36\) खींची गई है। समाधान क्षेत्र को मूल बिंदु की ओर और पहले चतुर्थांश में (त्रिभुज OAB का आंतरिक भाग और भुजाएँ) छायांकित किया गया है।
In simple words: To formulate, let x be grapes and y be peaches; the budget constraint is \(150x + 200y \le 1800\), simplifying to \(3x + 4y \le 36\). Graph this line and shade the region towards the origin in the first quadrant, as quantities cannot be negative.

🎯 Exam Tip: In word problems involving budget constraints, the inequality will typically be "less than or equal to" (≤). Always remember to include the non-negativity constraints (\(x \ge 0, y \ge 0\)) for real-world quantities, limiting the solution to the first quadrant.

Question 3. The Diet of the sick person must contain at least 4000 units of vitamin. Each unit of food F₁ contains 200 units of vitamin, whereas each unit of food F2 contains 100 units of vitamins. Write an inequation to fulfill a sick person's requirements and represent the solution set graphically.
Answer: Solution:
Let the diet of the sick person contain, x units of food F₁ and y units of food F2.
Since each unit of food F₁ contains 200 units of vitamins.
∴ x units of food F₁ contain 200x units of vitamins.
Also, each unit of food F2 contains 100 units of vitamins.
y units of food F2 contain 100y units of vitamins.
Now, Diet for a sick person must contain at least 4000 units of vitamins.
∴ he must take food F₁ and F2 in such away that total vitamins must be greater than or equal to 4000.
∴ required inequation is 200x + 100y ≥ 4000
i.e., 2x + y ≥ 40
Also x and y cannot be negative.
∴ x ≥ 0, y ≥ 0
Corresponding equation is 2x + y = 40
∴ \( \frac{2x}{40} + \frac{y}{40} = \frac{40}{40} \)
\( \implies \frac{x}{20} + \frac{y}{40} = 1 \)
∴ intersection of line with X-axis is A(20, 0)
intersection of line with Y-axis is B(0, 40)
Origin test:
Substituting x = 0, y = 0 in inequation, we get
2(0) + (0) ≥ 40
∴ 0 ≥ 40 which is false
∴ all the points on the non origin side of the line and points on the line satisfy the inequation.
Also, x ≥ 0, y ≥ 0
∴ the solution set is as shown in the figure.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक दो-आयामी तल में असमानता \(2x + y \ge 40\) का ग्राफिक समाधान प्रस्तुत करता है, साथ ही \(x \ge 0\) और \(y \ge 0\) की शर्तों को भी दर्शाता है। इसमें X-अक्ष को A(20, 0) पर और Y-अक्ष को B(0, 40) पर काटने वाली रेखा \(2x + y = 40\) खींची गई है। समाधान क्षेत्र को मूल बिंदु के विपरीत दिशा में और पहले चतुर्थांश में (रेखा के ऊपर का क्षेत्र) छायांकित किया गया है।
In simple words: Let x be units of F1 and y be units of F2. The vitamin requirement translates to \(200x + 100y \ge 4000\), which simplifies to \(2x + y \ge 40\). Graph the line \(2x + y = 40\) and, since \(0 \ge 40\) is false, shade the region away from the origin in the first quadrant.

🎯 Exam Tip: In minimum requirement problems, the inequality will typically be "greater than or equal to" (≥). Always simplify the inequality coefficients to their lowest possible integers for easier calculation and graphing.