Linear Inequalities

Exercise 5.1 • Scroll Horizontally for Full Steps
1. 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: $x < \frac{100}{24}$
Step 2 Simplify: $x < \frac{25}{6}$
Decimal $x < 4.166...$
(i) Natural Numbers: $\{1, 2, 3, 4\}$
(ii) Integers: $\{…, -3, -2, -1, 0, 1, 2, 3, 4\}$
2. Solve $-12x > 30$, when (i) x is a natural number, (ii) x is an integer.
Given $-12x > 30$
Step 1 Divide by -12 (Reverse Sign): $x < \frac{30}{-12}$
Result $x < -2.5$
(i) Natural: No Solution (None are < -2.5)
(ii) Integers: $\{…, -5, -4, -3\}$
3. Solve $5x – 3 < 7$, when (i) x is integer, (ii) x is real number.
Add 3 $5x < 7 + 3$
Simplify $5x < 10$
Divide 5 $x < 2$
(i) Integers: $\{…, -2, -1, 0, 1\}$
(ii) Real: Interval $(-\infty, 2)$
4. Solve $3x + 8 > 2$, when (i) x is integer, (ii) x is real number.
Subtract 8 $3x > 2 – 8$
Simplify $3x > -6$
Divide 3 $x > -2$
(i) Integers: $\{-1, 0, 1, 2, …\}$
(ii) Real: Interval $(-2, \infty)$
5. Solve $4x + 3 < 5x + 7$ for real x.
Group x $4x – 5x < 7 - 3$
Simplify $-x < 4$
Multiply -1 $x > -4$ (Sign Flips)
Solution: $(-4, \infty)$
6. Solve $3x – 7 > 5x – 1$ for real x.
Group x $3x – 5x > -1 + 7$
Simplify $-2x > 6$
Divide -2 $x < -3$ (Sign Flips)
Solution: $(-\infty, -3)$
7. Solve $3(x – 1) \le 2(x – 3)$ for real x.
Expand $3x – 3 \le 2x – 6$
Group x $3x – 2x \le -6 + 3$
Solve $x \le -3$
Solution: $(-\infty, -3]$
8. Solve $3(2 – x) \ge 2(1 – x)$ for real x.
Expand $6 – 3x \ge 2 – 2x$
Group x $-3x + 2x \ge 2 – 6$
Simplify $-x \ge -4$
Result $x \le 4$
Solution: $(-\infty, 4]$
9. Solve $x + \frac{x}{2} + \frac{x}{3} < 11$ for real x.
LCM (6) $\frac{6x + 3x + 2x}{6} < 11$
Simplify $\frac{11x}{6} < 11$
Multiply 6 $11x < 66$
Result $x < 6$
Solution: $(-\infty, 6)$
10. Solve $\frac{x}{3} > \frac{x}{2} + 1$ for real x.
Multiply 6 $2x > 3x + 6$
Group x $2x – 3x > 6$
Simplify $-x > 6$
Result $x < -6$
Solution: $(-\infty, -6)$
11. Solve $\frac{3(x-2)}{5} \le \frac{5(2-x)}{3}$ for real x.
Cross Mult $3 \cdot 3(x-2) \le 5 \cdot 5(2-x)$
Expand $9x – 18 \le 50 – 25x$
Group x $9x + 25x \le 50 + 18$
Simplify $34x \le 68$
Result $x \le 2$
Solution: $(-\infty, 2]$
12. Solve $\frac{1}{2}(\frac{3x}{5} + 4) \ge \frac{1}{3}(x-6)$ for real x.
Multiply 6 $3(\frac{3x}{5} + 4) \ge 2(x-6)$
Expand $\frac{9x}{5} + 12 \ge 2x – 12$
Group $12 + 12 \ge 2x – \frac{9x}{5}$
Common Den $24 \ge \frac{10x – 9x}{5}$
Result $120 \ge x$ (or $x \le 120$)
Solution: $(-\infty, 120]$
13. Solve $2(2x + 3) – 10 < 6(x - 2)$ for real x.
Expand $4x + 6 – 10 < 6x - 12$
Combine $4x – 4 < 6x - 12$
Group x $4x – 6x < -12 + 4$
Simplify $-2x < -8$
Result $x > 4$ (Sign Flip)
Solution: $(4, \infty)$
14. Solve $37 – (3x + 5) \ge 9x – 8(x – 3)$ for real x.
Expand $37 – 3x – 5 \ge 9x – 8x + 24$
Combine $32 – 3x \ge x + 24$
Group x $32 – 24 \ge x + 3x$
Simplify $8 \ge 4x$
Result $2 \ge x$ (or $x \le 2$)
Solution: $(-\infty, 2]$
15. Solve $\frac{x}{4} < \frac{5x-2}{3} - \frac{7x-3}{5}$ for real x.
RHS LCM $\frac{5(5x-2) – 3(7x-3)}{15}$
Expand $\frac{25x – 10 – 21x + 9}{15} = \frac{4x – 1}{15}$
Inequality $\frac{x}{4} < \frac{4x - 1}{15}$
Cross Mult $15x < 4(4x - 1) \Rightarrow 15x < 16x - 4$
Solve $4 < 16x - 15x \Rightarrow 4 < x$
Solution: $(4, \infty)$
16. Solve $\frac{2x-1}{3} \ge \frac{3x-2}{4} – \frac{2-x}{5}$ for real x.
RHS LCM $\frac{5(3x-2) – 4(2-x)}{20}$
Expand $\frac{15x – 10 – 8 + 4x}{20} = \frac{19x – 18}{20}$
Cross Mult $20(2x-1) \ge 3(19x-18)$
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 $-5x < 5$
Result $x > -1$ (Sign Flip)
-1
Solution: $(-1, \infty)$
20. Solve $\frac{x}{2} \ge \frac{5x-2}{3} – \frac{7x-3}{5}$ and graph.
Algebra Same as Q15 (LCM 30).
Equation $15x \ge 8x – 2$
Result $7x \ge -2 \Rightarrow x \ge -\frac{2}{7}$
-2/7
Solution: $[-\frac{2}{7}, \infty)$
21. Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
Given Marks: 70, 75. Let 3rd test be $x$.
Avg Formula $\frac{70 + 75 + x}{3} \ge 60$
Mult 3 $145 + x \ge 180$
Result $x \ge 35$
Minimum Marks = 35
22. To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations. If Sunita’s marks in first four are 87, 92, 94 and 95, find minimum marks for fifth exam.
Given Marks: 87, 92, 94, 95. Let 5th be $x$.
Avg $\ge$ 90 $\frac{87 + 92 + 94 + 95 + x}{5} \ge 90$
Mult 5 $\frac{368 + x}{5} \ge 90 \Rightarrow 368 + x \ge 450$
Result $x \ge 82$
Minimum Marks = 82
23. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
Setup Let integers be $x, x+2$.
Cond 1 $x < 10$ (and $x+2 < 10 \Rightarrow x < 8$)
Cond 2 $x + (x+2) > 11 \Rightarrow 2x > 9 \Rightarrow x > 4.5$
Range Odd integers between 4.5 and 8 are 5 and 7.
Pairs: $(5,7), (7,9)$
24. Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
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 Even integers between 5 and 10.5 are 6, 8, 10.
Pairs: $(6,8), (8,10), (10,12)$
25. The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
Sides Shortest=$x$, Longest=$3x$, Third=$3x-2$
Perimeter $x + 3x + (3x-2) \ge 61$
Solve $7x – 2 \ge 61 \Rightarrow 7x \ge 63$
Result $x \ge 9$
Minimum length of shortest side = 9 cm
26. A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?
Define Shortest=$x$, Second=$x+3$, Third=$2x$
Total Len $x + (x+3) + 2x \le 91 \Rightarrow 4x \le 88 \Rightarrow x \le 22$
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
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