04. Linear Inequations (in One Variable) Previous Year Questions

Multiply by −3 (reversing inequality signs): 0 > x > −6 ⇔ −6 < x < 0. Since x ∈ Z (integers), the solution set is {−5, −4, −3, −2, −1}. [1 Mark]

Part 1: 2x − 5/3 < 3x/5 + 10 ⇒ 30x − 25 < 9x + 150 ⇒ 21x < 175 ⇒ x < 25/3 (8⅓). [1 Mark]
Part 2: 3x/5 + 10 ≤ 4x/5 + 11 ⇒ 3x + 50 ≤ 4x + 55 ⇒ x ≥ −5. [1 Mark]
Combined Solution Set: {x : −5 ≤ x < 25/3, x ∈ R}.
Number Line Representation: A solid darkened circle at −5, heavy line shaded continuously up to 25/3 (8.33), with a hollow open circle at 25/3. [1 Mark]

Part 1: −3 + x ≤ ^7x/₂ + 2 ⇒ −6 + 2x ≤ 7x + 4 ⇒ 5x ≥ −10 ⇒ x ≥ −2. [1 Mark]
Part 2: ^7x/₂ + 2 < 8 + 2x ⇒ 7x + 4 < 16 + 4x ⇒ 3x < 12 ⇒ x < 4. [1 Mark]
Combining: −2 ≤ x < 4. Since x ∈ I:
Solution Set = {−2, −1, 0, 1, 2, 3}. Correct solid dots plotted on number line. [1 Mark]

$2x + 4 \le 14 \implies 2x \le 10 \implies x \le 5$.
Since $x \in W$ (set of whole numbers, which starts from 0):
Solution set = $\{0, 1, 2, 3, 4, 5\}$.
Correct Option: (b) [1 Mark]

Part 1 Split into two simultaneous inequations:
Left part: $-5(x - 9) \ge 17 - 9x$
$-5x + 45 \ge 17 - 9x \implies -5x + 9x \ge 17 - 45 \implies 4x \ge -28 \implies x \ge -7$.1 Mark

Right part: $17 - 9x > x + 2$
$-9x - x > 2 - 17 \implies -10x > -15 \implies x < \frac{15}{10} = 1.5$.1 Mark

Part 2 Combining solutions for $x \in R$:
$$-7 \le x < 1.5$$ Solution Set: $\{x : -7 \le x < 1.5, x \in R\}$.
Number Line: Closed dot at $-7$, thick bold segment up to $1.5$, hollow circle at $1.5$.1 Mark

Part 1 $\frac{3x}{5} + 2 < x + 4 \implies 2 - 4 < x - \frac{3x}{5} \implies -2 < \frac{2x}{5} \implies -10 < 2x \implies \mathbf{x > -5}$. [1 Mark]
Part 2 $x + 4 \le \frac{x}{2} + 5 \implies x - \frac{x}{2} \le 5 - 4 \implies \frac{x}{2} \le 1 \implies \mathbf{x \le 2}$. [1 Mark]
Step 3 Combining both parts: $-5 < x \le 2, \quad x \in \mathbb{R}$.
Solution set $= \{x : -5 < x \le 2, x \in \mathbb{R}\}$. Correct plot with hollow circle at $-5$ and solid dot at $2$. [1 Mark]

Step 1 Split the inequation into two linear parts:
Part 1: $11x - 4 < 15x + 4$
$\implies -4 - 4 < 15x - 11x \implies -8 < 4x \implies x > -2$. [1 Mark]
Part 2: $15x + 4 \le 13x + 14$
$\implies 15x - 13x \le 14 - 4 \implies 2x \le 10 \implies x \le 5$. [1 Mark]
Step 2 Combine inequalities: $-2 < x \le 5$.
Since $x \in W$ (set of whole numbers $\{0, 1, 2, 3, \dots\}$):
Solution Set: $\{0, 1, 2, 3, 4, 5\}$.
Step 3 Graph the discrete elements with solid dark dots on the real number line at $0, 1, 2, 3, 4, 5$. [1 Mark]

Step 1 Solve linear inequation in two parts:
Part 1: $3x + \frac{14}{3} > \frac{4x}{3} - 2$
$\implies 3x - \frac{4x}{3} > -2 - \frac{14}{3} \implies \frac{5x}{3} > -\frac{20}{3} \implies 5x > -20 \implies x > -4$. [1 Mark]
Part 2: $\frac{4x}{3} - 2 \ge 3x - 4$
$\implies -2 + 4 \ge 3x - \frac{4x}{3} \implies 2 \ge \frac{5x}{3} \implies 5x \le 6 \implies x \le \frac{6}{5} = 1.2$. [1 Mark]
Step 2 Combine inequalities:
$\mathbf{\text{Solution Set: } \{x : x \in \mathbb{R}, -4 < x \le 1.2\}}$.
Graph with open circle at $-4$, filled solid dot at $1.2$, and thickened segment between them. [1 Mark]

Part 1 First inequality:
$$-2 + 10x \le 13x + 10 \implies -2 - 10 \le 13x - 10x$$ $$-12 \le 3x \implies x \ge -4$$ [1 Mark]
Part 2 Second inequality:
$$13x + 10 < 24 + 10x \implies 13x - 10x < 24 - 10$$ $$3x < 14 \implies x < \frac{14}{3} \approx 4.67$$ [1 Mark]
Solution Set Combining both: $-4 \le x < \frac{14}{3}$.
Since $x \in \mathbb{Z}$ (integers):
$$\text{Solution Set} = \mathbf{\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}}$$ [0.5 Mark]
Number line Represented by solid darkened dots on integers from $-4$ to $4$ inclusive. [0.5 Mark]
Answer: Solution Set $= \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$.

Step 1 Add $\frac{1}{2}$ throughout the inequation:
$$-8.5 + 0.5 < -4x \le 7.5 + 0.5 \implies -8 < -4x \le 8$$ [1 Mark]
Step 2 Divide throughout by $-4$ (reversing the inequality signs):
$$\frac{-8}{-4} > x \ge \frac{8}{-4} \implies 2 > x \ge -2 \iff -2 \le x < 2$$ [1 Mark]
Step 3: Solution Set Since $x \in I$ (integers):
$$\text{Solution Set} = \mathbf{\{-2, -1, 0, 1\}}$$ [0.5 Mark]
Represented by solid filled dots on $-2, -1, 0, 1$ on the number line. [0.5 Mark]
Answer: Solution Set $= \{-2, -1, 0, 1\}$.