Inequalities Worksheet

Solutions

1 Solve:

Inequality Exercise

2Solve:

4x2 − 4x + 1 ≤ 0

3Solve:

Inequality Exercise

4Calculate the values of k for which the roots of the equation x2 − 6x + k = 0 are two real and distinct numbers.

5Solve:

1Inequality Exercise

2Inequality Exercise

3Inequality Exercise

6Solve:

Inequality Exercise

7Solve:

1inecuación

2x4 − 25x2 − 144 < 0

3x4 − 16x2 − 225 ≥ 0  


1

Solve:

Inequality Exercise

Inequality Operations

Inequality Operations

Inequality Operations

Inequality Operations

Inequality Graph

Inequality Solution


2

Solve:

4x2 − 4x + 1 ≤ 0

4x2 − 4x + 1 = 0

Inequality Operations

Inequality Solution


3

Solve:

Inequality Exercise

Inequality Operations     

    The numerator is always positive.

Inequality Operations    

The denominator cannot be zero.

Inequality Operations

Therefore, the original inequality will be equivalent to:

x2 − 4 > 0

Inequality Graph

(−-∞ , −2) Unión (2, +∞)


4

Calculate the values of k for which the roots of the equation x2 − 6x + k = 0 are two real and distinct numbers.

(−6)2 − 4k > 0

36 − 4k > 0          − 4k > − 36        k < 9

Inequality Graph

(−∞, 9)


5

Solve:

1Inequality Exercise

x = 4

y = 2

Inequality Graph

2Inequality Exercise

x + y = 0        (0, 0)     (1, -1)

2 + 2 ≥ 0

Inequality Graph

2x − y = 0      (0, 0)     (1, 2)

2 ·2 − 2 ≥ 0

Inequality Graph

Inequality Graph

3Inequality Exercise

x + y = 0        (0, 0)     (1, -1)

2 + 2 ≥ 0

Inequality Graph

2x − y = 0      (0, 0)     (1, 2)

2 ·2 − 2 ≥ 0

Inequality Graph

2 ≤ 6

     Inequality Graph

Inequality Graph


6

Solve:

Inequality Exercise

(x +1) · 10 + x ≤ 6 (2x + 1)

10x + 10 + x ≤ 12 x + 6

10 x + x - 12x ≤ 6 - 10

−x − 4       x ≥ 4

Inequality Operations

Inequality Operations

Inequality Operations

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Inequality Graph

[4, 7)


7

Solve:

1inecuación

inecuación

As the first factor is always positive, consider the sign of the 2nd factor.

Inequality Operations

solución

Inequality Graph

P(−17) = (−17) 2 + 12 · 17 − 64 > 0

P(0) = 02 + 12 · 0 − 64 < 0

P(5) = 5 2 + 12 · 5 − 64 > 0

Inequality Graph

(-∞, −16] Unión [4, ∞)

2x4 − 25x2 − 144 < 0

x4 − 25x2 − 144 = 0

Inequality Operations

Inequality Operations

Inequality Operations

Inequality Operations

Inequality Graph

(−4, −3) Unión (3, 4) .

3x4 − 16x2 − 225 ≥ 0 

x4 − 16x2 − 225 = 0 

Inequality Operations

Inequality Operations

Inequality Operations

Inequality Operations

(x2 - 25) · (x2 + 9) ≥ 0

The second factor is always positive and nonzero, therefor, only consider the sign of the 1st factor.

(x2 − 25) ≥ 0

Inequality Graph

(-∞, −5] Unión [5, +∞)