Questions & Answers: "Function - Linear inequality"

To solve the inequality (x / 4) < (5x - 2) / 3 - (7x - 3) / 5, first find the LCM of the denominators on the RHS (3 and 5), which is 15. Simplify the RHS: [5(5x - 2) - 3(7x - 3)] / 15 = (25x - 10 - 21x + 9) / 15 = (4x - 1) / 15. Now we have x / 4 < (4x - 1) / 15. Multiply both sides by 60 (the LCM of 4 and 15) to clear fractions: 15x 15x < 16x - 4. Subtracting 16x from both sides gives -x 4. In interval notation, this is (4, ∞). Option B is incorrect as it represents x < 4. Options C and D are incorrect because the original inequality was strict (<), meaning the endpoint 4 is not included.

The problem states the temperature must be between 68°F and 77°F, which gives the double inequality: 68 < F < 77. Substitute the formula for F: 68 < (9/5)C + 32 < 77. Subtract 32 from all parts: 68 - 32 < (9/5)C 36 < (9/5)C < 45. To solve for C, multiply the entire inequality by 5/9: (36 * 5/9) < C 4 * 5 < C 20 < C < 25. The interval is (20, 25). Option B is wrong because 'between' implies strict inequality (open brackets). Options C and D are numerically incorrect based on the calculation.

Solve each inequality separately. For the first: 3x - 7 3x - x 2x x 11 - 1 ≤ 5x => 10 ≤ 5x => 2 ≤ x. This means x must be greater than or equal to 2 AND strictly less than 6. Combining these, we get 2 ≤ x < 6. In interval notation, this is written as [2, 6). Option B reverses the inclusion of endpoints. Option C incorrectly includes 6. Option D incorrectly excludes 2.

The absolute value inequality |x - a| ≤ r is equivalent to a - r ≤ x ≤ a + r. Here, a = 1 and r = 2. So, 1 - 2 ≤ x ≤ 1 + 2, which gives -1 ≤ x ≤ 3. In interval notation, this is [-1, 3]. Option 1 is incorrect because it uses open intervals. Options 3 and 4 are incorrect due to calculation errors in applying the boundary values.

Expanding both sides: 6 - 3x ≥ 2 - 2x. Rearranging terms: 6 - 2 ≥ 3x - 2x, which simplifies to 4 ≥ x, or x ≤ 4. Option 2 is the reverse. Option 3 has a sign error. Option 4 is incorrect because it uses a strict inequality and the wrong direction.

The critical points are x = 2 and x = -5. These points divide the number line into three intervals: (-∞, -5), (-5, 2), and (2, ∞). Testing a point in each: for x=3, (3-2)/(3+5) = 1/8 > 0 (Positive); for x=0, (0-2)/(0+5) = -2/5 0 (Positive). Since the inequality is strictly greater than 0, we exclude endpoints where the expression is zero or undefined. Thus, the solution is (-∞, -5) ∪ (2, ∞). Option 1 is the region where it is negative. Options 3 and 4 incorrectly include endpoints.

To solve 1/x 0, then 1 x > 1/2. Case 2: If x 2x => 1/2 > x. Since we assumed x < 0, all x < 0 satisfy this. Thus, the solution is x 1/2. Option 2 misses the negative values. Option 3 is where 1/x > 2. Option 4 is incorrect because it includes values like 0.1 where 1/0.1=10, which is not < 2.

The inequalities x ≥ 0 and y ≥ 0 restrict the solution to the first quadrant. The inequality x + y ≤ 4 represents the region below and on the line passing through (4,0) and (0,4). The intersection of these three constraints forms a closed triangular region with vertices (0,0), (4,0), and (0,4). It is bounded, making Option 2 wrong. It's not a rectangle (Option 3) and not in the third quadrant (Option 4).

Multiply the entire inequality by -2. Since -2 is negative, the inequality signs reverse: -12 * (-2) > 3x - 5 ≥ 6 * (-2). This gives 24 > 3x - 5 ≥ -12. Adding 5 to all parts: 29 > 3x ≥ -7. Dividing by 3: 29/3 > x ≥ -7/3... wait, let's re-calculate. -12 3x - 5 ≥ -12. Add 5: 29 > 3x ≥ -7. Divide by 3: 29/3 > x ≥ -7/3. This translates to [-7/3, 29/3). Looking at options, let's assume the question meant 3x-5 without division or check formatting. Re-evaluating: if 3x-5/(-2), result is [-7/3, 29/3). If the question was -12 < 3x-5 <= 6, results differ. Given typical NCERT options, if the denominator was 2 (positive), the signs wouldn't flip. However, for the provided correct option logic: -12 24 > 3x-5 => 29 > 3x => x < 29/3. And (3x-5)/-2 3x-5 >= -12 => 3x >= -7 => x >= -7/3. Thus [-7/3, 29/3). (Option B adjusted for integer simplification in typical tests). Correction: If 3x-5 was 4x-something, numbers would be cleaner. Based on the logic, x is in [-7/3, 29/3).

To solve 4x + 3 < 5x + 7, we rearrange terms: 4x - 5x < 7 - 3, which simplifies to -x -4. In interval notation, this is represented as (-4, ∞). Option 2 is incorrect because it suggests x < -4. Option 3 is incorrect because the inequality is strict ('<'), meaning -4 is not included. Option 4 is incorrect due to a sign error.

Solve each inequality separately: 1) 2x + 1 > 3 => 2x > 2 => x > 1. 2) 3x - 5 3x x 1 and x < 6 is 1 < x < 6, or (1, 6). Option 2 is incorrect because the inequalities are strict. Options 3 and 4 only represent one half of the system's requirements.

Let the score in the fourth test be x. The average is (50 + 48 + 72 + x) / 4. We need this to be ≥ 60. So, (170 + x) / 4 ≥ 60. Multiplying by 4: 170 + x ≥ 240. Subtracting 170: x ≥ 70. Thus, the minimum score required is 70. Option 1 is too low (average would be 57.5). Options 3 and 4 are mathematically incorrect relative to the 'minimum' requirement.

The inequality |x| > a means x > a or x 5 or 2x - 3 8 => x > 4. Solving the second: 2x x < -1. Combining these, we get x 4, which is (-∞, -1) ∪ (4, ∞). Option 1 is the solution for the reverse inequality (|2x-3| < 5). Option 4 is incomplete.