Linear Inequalities
Exercise 5.1 Solutions1. Solve $24x < 100$, when (i) x is a natural number, (ii) x is an integer.
Given
$24x < 100$
Step 1
Divide both sides by 24:
$x < \frac{100}{24}$
$x < \frac{100}{24}$
Step 2
Simplify the fraction:
$x < \frac{25}{6}$
$x < \frac{25}{6}$
Decimal
$x < 4.166...$
(i) If x is a Natural Number:
$\{1, 2, 3, 4\}$
(ii) If x is an Integer:
$\{…, -3, -2, -1, 0, 1, 2, 3, 4\}$
2. Solve $-12x > 30$, when (i) x is a natural number, (ii) x is an integer.
Given
$-12x > 30$
Step 1
Divide by -12:
$x < \frac{30}{-12}$ ⚠ Warning: Dividing by negative reverses the inequality sign!
$x < \frac{30}{-12}$ ⚠ Warning: Dividing by negative reverses the inequality sign!
Step 2
$x < -\frac{5}{2}$ or $x < -2.5$
(i) If x is a Natural Number:
No Solution (Since natural numbers are positive integers {1, 2, 3…}, none are less than -2.5)
(ii) If x is an Integer:
$\{…, -5, -4, -3\}$
3. Solve $5x – 3 < 7$, when (i) x is an integer, (ii) x is a real number.
Given
$5x – 3 < 7$
Step 1
$5x < 7 + 3$
Step 2
$5x < 10$
Step 3
$x < 2$
(i) Integer:
$\{…, -2, -1, 0, 1\}$
(ii) Real Number:
Interval $(-\infty, 2)$
4. Solve $3x + 8 > 2$, when (i) x is an integer, (ii) x is a real number.
Given
$3x + 8 > 2$
Step 1
$3x > 2 – 8$
Step 2
$3x > -6$
Step 3
$x > -2$
(i) Integer:
$\{-1, 0, 1, 2, …\}$
(ii) Real Number:
Interval $(-2, \infty)$
5. Solve $4x + 3 < 5x + 7$ for real x.
Given
$4x + 3 < 5x + 7$
Step 1
Move terms with x to LHS:
$4x – 5x < 7 - 3$
$4x – 5x < 7 - 3$
Step 2
$-x < 4$
Step 3
Multiply by -1:
$x > -4$ ⚠ Sign flips!
$x > -4$ ⚠ Sign flips!
Solution: $(-4, \infty)$
6. Solve $3x – 7 > 5x – 1$ for real x.
Step 1
$3x – 5x > -1 + 7$
Step 2
$-2x > 6$
Step 3
Divide by -2:
$x < -3$
$x < -3$
Solution: $(-\infty, -3)$
7. Solve $3(x – 1) \le 2(x – 3)$ for real x.
Expand
$3x – 3 \le 2x – 6$
Step 1
$3x – 2x \le -6 + 3$
Step 2
$x \le -3$
Solution: $(-\infty, -3]$
8. Solve $3(2 – x) \ge 2(1 – x)$ for real x.
Expand
$6 – 3x \ge 2 – 2x$
Step 1
$-3x + 2x \ge 2 – 6$
Step 2
$-x \ge -4$
Step 3
$x \le 4$
Solution: $(-\infty, 4]$
9. Solve $x + \frac{x}{2} + \frac{x}{3} < 11$ for real x.
Given
$x + \frac{x}{2} + \frac{x}{3} < 11$
Step 1
Take LCM of 2 and 3 (which is 6):
$\frac{6x + 3x + 2x}{6} < 11$
$\frac{6x + 3x + 2x}{6} < 11$
Step 2
$\frac{11x}{6} < 11$
Step 3
$11x < 66$
Step 4
$x < 6$
Solution: $(-\infty, 6)$
10. Solve $\frac{x}{3} > \frac{x}{2} + 1$ for real x.
Step 1
Multiply entire equation by 6 (LCM):
$2x > 3x + 6$
$2x > 3x + 6$
Step 2
$2x – 3x > 6$
Step 3
$-x > 6$
Step 4
$x < -6$
Solution: $(-\infty, -6)$
11. Solve $\frac{3(x-2)}{5} \le \frac{5(2-x)}{3}$ for real x.
Step 1
Cross multiply (positive denominators):
$3 \cdot 3(x-2) \le 5 \cdot 5(2-x)$
$3 \cdot 3(x-2) \le 5 \cdot 5(2-x)$
Step 2
$9(x-2) \le 25(2-x)$
Step 3
$9x – 18 \le 50 – 25x$
Step 4
$9x + 25x \le 50 + 18$
Step 5
$34x \le 68 \Rightarrow x \le 2$
Solution: $(-\infty, 2]$
12. Solve $\frac{1}{2}(\frac{3x}{5} + 4) \ge \frac{1}{3}(x-6)$ for real x.
Step 1
Multiply by 6 to clear outer fractions:
$3(\frac{3x}{5} + 4) \ge 2(x-6)$
$3(\frac{3x}{5} + 4) \ge 2(x-6)$
Step 2
Expand terms:
$\frac{9x}{5} + 12 \ge 2x – 12$
$\frac{9x}{5} + 12 \ge 2x – 12$
Step 3
Group constants:
$12 + 12 \ge 2x – \frac{9x}{5}$
$12 + 12 \ge 2x – \frac{9x}{5}$
Step 4
$24 \ge \frac{10x – 9x}{5} \Rightarrow 24 \ge \frac{x}{5}$
Step 5
$120 \ge x$ (or $x \le 120$)
Solution: $(-\infty, 120]$
13. Solve $2(2x + 3) – 10 < 6(x - 2)$ for real x.
Expand
$4x + 6 – 10 < 6x - 12$
Step 1
$4x – 4 < 6x - 12$
Step 2
$4x – 6x < -12 + 4$
Step 3
$-2x < -8$
Step 4
$x > 4$ (Sign Flip)
Solution: $(4, \infty)$
14. Solve $37 – (3x + 5) \ge 9x – 8(x – 3)$ for real x.
Expand
$37 – 3x – 5 \ge 9x – 8x + 24$
Step 1
$32 – 3x \ge x + 24$
Step 2
$32 – 24 \ge x + 3x$
Step 3
$8 \ge 4x \Rightarrow 2 \ge x$
Solution: $(-\infty, 2]$
15. Solve $\frac{x}{4} < \frac{5x-2}{3} - \frac{7x-3}{5}$ for real x.
RHS
LCM of 3 and 5 is 15.
$\frac{5(5x-2) – 3(7x-3)}{15}$
$\frac{5(5x-2) – 3(7x-3)}{15}$
Expand
$\frac{25x-10-21x+9}{15} = \frac{4x-1}{15}$
Combine
$\frac{x}{4} < \frac{4x-1}{15}$
Cross
$15x < 4(4x-1) \Rightarrow 15x < 16x - 4$
Result
$4 < x$ (or $x > 4$)
Solution: $(4, \infty)$
16. Solve $\frac{2x-1}{3} \ge \frac{3x-2}{4} – \frac{2-x}{5}$ for real x.
RHS
LCM 20:
$\frac{5(3x-2) – 4(2-x)}{20}$
$\frac{5(3x-2) – 4(2-x)}{20}$
Expand
$\frac{15x-10-8+4x}{20} = \frac{19x-18}{20}$
Cross
$20(2x-1) \ge 3(19x-18)$
Simplify
$40x – 20 \ge 57x – 54$
Solve
$54 – 20 \ge 57x – 40x \Rightarrow 34 \ge 17x \Rightarrow 2 \ge x$
Solution: $(-\infty, 2]$
17. Solve $3x – 2 < 2x + 1$ and graph.
Step 1
$3x – 2x < 1 + 2$
Result
$x < 3$
3
Solution: $(-\infty, 3)$
18. Solve $5x – 3 > 3x – 5$ and graph.
Step 1
$5x – 3x > -5 + 3$
Step 2
$2x > -2$
Result
$x > -1$
-1
Solution: $(-1, \infty)$
19. Solve $3(1 – x) < 2(x + 4)$ and graph.
Expand
$3 – 3x < 2x + 8$
Step 1
$-3x – 2x < 8 - 3$
Step 2
$-5x < 5$
Step 3
$x > -1$ (Sign Flip)
-1
Solution: $(-1, \infty)$
20. Solve $\frac{x}{2} \ge \frac{5x-2}{3} – \frac{7x-3}{5}$ and graph.
Setup
Using previous algebraic result (Q16 logic):
Result
$x \ge -\frac{2}{7}$
-2/7
Solution: $[-\frac{2}{7}, \infty)$
21. Ravi obtained 70 and 75 marks. Find min marks for average of 60.
Given
Marks: 70, 75. Let 3rd test be $x$.
Equation
$\frac{70 + 75 + x}{3} \ge 60$
Step 1
$145 + x \ge 180$
Step 2
$x \ge 35$
Minimum marks = 35
22. Sunita’s Grade A (Average $\ge$ 90). Marks: 87, 92, 94, 95.
Given
Marks: 87, 92, 94, 95. Let 5th be $x$.
Equation
$\frac{87+92+94+95+x}{5} \ge 90$
Step 1
$\frac{368+x}{5} \ge 90$
Step 2
$368+x \ge 450$
Step 3
$x \ge 82$
Minimum marks = 82
23. Consecutive odd integers: Sum > 11, Both < 10.
Setup
Let integers be $x, x+2$.
Cond 1
$x < 10$
Cond 2
$x + (x+2) > 11 \Rightarrow 2x > 9 \Rightarrow x > 4.5$
Range
$x$ is an odd integer between 4.5 and 8 (since $x+2 < 10 \Rightarrow x < 8$).
Values
$x$ can be 5 or 7.
Pairs: $(5,7), (7,9)$
24. Consecutive even integers: Sum < 23, Both > 5.
Setup
Let integers be $x, x+2$.
Cond 1
$x > 5$
Cond 2
$x + (x+2) < 23 \Rightarrow 2x < 21 \Rightarrow x < 10.5$
Range
$x$ is an even integer between 5 and 10.5.
Values
$x$ can be 6, 8, 10.
Pairs: $(6,8), (8,10), (10,12)$
25. Triangle Sides: Longest is 3x shortest, Third is 2cm less than longest. Perimeter $\ge$ 61.
Define
Shortest = $x$. Longest = $3x$. Third = $3x-2$.
Inequality
Perimeter $\ge 61 \Rightarrow x + 3x + (3x-2) \ge 61$
Step 1
$7x – 2 \ge 61$
Step 2
$7x \ge 63$
Step 3
$x \ge 9$
Minimum length of shortest side = 9 cm
26. Board Cutting: Length 91cm. 2nd is 3cm longer than shortest. 3rd is 2x shortest.
Define
Shortest = $x$, Second = $x+3$, Third = $2x$.
Total
$x + (x+3) + 2x \le 91$
Solve 1
$4x + 3 \le 91 \Rightarrow 4x \le 88 \Rightarrow x \le 22$
Constraint
3rd $\ge$ 2nd + 5 $\Rightarrow 2x \ge (x+3) + 5$
Solve 2
$2x \ge x + 8 \Rightarrow x \ge 8$
Shortest board length: $8 \le x \le 22$ cm