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secondary 3 | A Maths
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Date Posted: 2 years ago
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|4 - 3x| ≥ |2x + 1|
Square both sides and remove the modulus signs.
(4 - 3x)² ≥ (2x + 1)²
Expand both sides,
16 - 24x + 9x² ≥ 4x² + 4x + 1
5x² - 28x + 15 ≥ 0
(5x - 3)(x - 5) ≥ 0
5x ≤ 3 or x ≥ 5
x ≤ 3/5
∴ x ≤ 3/5 or x ≥ 5
(It will help you to visualise when you try sketching out the quadratic curve (U-shaped, upward sloping) where the x-intercepts are 3/5 and 5.
Square both sides and remove the modulus signs.
(4 - 3x)² ≥ (2x + 1)²
Expand both sides,
16 - 24x + 9x² ≥ 4x² + 4x + 1
5x² - 28x + 15 ≥ 0
(5x - 3)(x - 5) ≥ 0
5x ≤ 3 or x ≥ 5
x ≤ 3/5
∴ x ≤ 3/5 or x ≥ 5
(It will help you to visualise when you try sketching out the quadratic curve (U-shaped, upward sloping) where the x-intercepts are 3/5 and 5.
Alternatively,
|4 - 3x| ≥ |2x + 1|
Two possibilities :
①
4 - 3x ≥|2x + 1|
4 - 3x ≥ 2x + 1 or 2x + 1 ≥ -(4 - 3x)
3 ≥ 5x or 2x + 1 ≥ 3x - 4
3/5 ≥ x or 5 ≥ x
Overall for ① : x ≤ 3/5 since it guarantees that x ≤ 5 as well.
Or
②
4 - 3x ≤ -|2x + 1|
3x - 4 ≥|2x + 1|
3x - 4 ≥ 2x + 1 or 2x + 1 ≥ -(3x - 4)
x ≥ 5 or 2x + 1 ≥ 4 - 3x
□□□□□5x ≥ 3
□□□□□x ≥ 3/5
Overall for ② : x ≥ 5 since it guarantees that x ≥ 3/5 as well.
Putting the two together,
∴ x ≤ 3/5 or x ≥ 5
|4 - 3x| ≥ |2x + 1|
Two possibilities :
①
4 - 3x ≥|2x + 1|
4 - 3x ≥ 2x + 1 or 2x + 1 ≥ -(4 - 3x)
3 ≥ 5x or 2x + 1 ≥ 3x - 4
3/5 ≥ x or 5 ≥ x
Overall for ① : x ≤ 3/5 since it guarantees that x ≤ 5 as well.
Or
②
4 - 3x ≤ -|2x + 1|
3x - 4 ≥|2x + 1|
3x - 4 ≥ 2x + 1 or 2x + 1 ≥ -(3x - 4)
x ≥ 5 or 2x + 1 ≥ 4 - 3x
□□□□□5x ≥ 3
□□□□□x ≥ 3/5
Overall for ② : x ≥ 5 since it guarantees that x ≥ 3/5 as well.
Putting the two together,
∴ x ≤ 3/5 or x ≥ 5
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