Linear Inequalities — Practice

Question 1

Solve -4x + 12 > 0 for real x.

Accepted Answer

-4x > -12. Dividing by -4 flips the sign: x < 3. Answer: The solution is x < 3, i.e. the interval (-infinity, 3).

Question 2

Solve 3x - 5 < x + 7 for real x, and write the solution in interval notation.

Accepted Answer

3x - x < 7 + 5, so 2x < 12, giving x < 6. Answer: The solution is x < 6, i.e. the interval (-infinity, 6).

Question 3

Solve 2(x-1) >= x+3 and describe how the solution is shown on a number line.

Accepted Answer

2x - 2 >= x + 3, so x >= 5. On a number line this is a filled (dark) circle at 5, since the inequality is non-strict, with shading to the right. Answer: The solution is x >= 5, shown with a filled circle at 5 and shading extending rightward.

Question 4

Solve -5 <= 2x - 3 < 9 for real x.

Accepted Answer

Add 3 across all three parts: -2 <= 2x < 12. Divide by 2: -1 <= x < 6. Answer: The solution is -1 <= x < 6, i.e. the interval [-1, 6).

Question 5

Solve the system: 2x - 3 < 7 and 3x + 1 >= 4.

Accepted Answer

First: 2x < 10, so x < 5. Second: 3x >= 3, so x >= 1. Intersecting both: 1 <= x < 5. Answer: The solution is 1 <= x < 5, i.e. the interval [1, 5).

Question 6

A student scored 55 and 68 in the first two unit tests. Find the minimum marks needed in the third test for an average of at least 65.

Accepted Answer

Let x be the third score. (55+68+x)/3 >= 65, so 123+x >= 195, giving x >= 72. Answer: The student needs a minimum of 72 marks in the third test.

Question 7

Find all pairs of consecutive even positive integers, both larger than 8, whose sum is less than 40.

Accepted Answer

Let x be the smaller even integer, so the other is x+2. Then x > 8 and x + (x+2) < 40, i.e. 2x < 38, so x < 19. Combining: 8 < x < 19, and since x is even, x can be 10, 12, 14, 16, or 18. Answer: The pairs are (10,12), (12,14), (14,16), (16,18), and (18,20).

Question 8

A manufacturer has 400 litres of a 10% acid solution. How many litres of a 25% acid solution must be added so the resulting mixture is more than 12% but less than 15% acid?

Accepted Answer

Let x litres of 25% solution be added; total mixture = (x+400) litres, with acid content 0.25x+0.10(400) = 0.25x+40. Requiring this between 12% and 15% of the total: 0.25x+40 > 0.12(x+400) and 0.25x+40 < 0.15(x+400). The first gives 0.13x > 8, so x > 800/13 (approximately 61.5). The second gives 0.10x < 20, so x < 200. Answer: Between 800/13 litres (about 61.5 L) and 200 litres of the 25% solution must be added.

Question 9

A liquid must be kept between 20°C and 25°C. Find the corresponding Fahrenheit range, given C = (5/9)(F-32).

Accepted Answer

20 < (5/9)(F-32) < 25. Multiplying through by 9/5: 36 < F-32 < 45. Adding 32: 68 < F < 77. Answer: The temperature must be kept between 68°F and 77°F.

Practice — Linear Inequalities

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities