Solve the inequality `3 - x > 1/|x - 4|` for `x`, expressing your answer in interval notation. (3 marks)
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Solve the inequality `3 - x > 1/|x - 4|` for `x`, expressing your answer in interval notation. (3 marks)
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`x ∈ (– oo, (7 – sqrt 5)/2)`
`3 – x > 1/|x – 4|`
`|x – 4| (3 – x) > 1`
`text(If)\ \ x – 4 > 0, x > 4`
`(x – 4) (3 – x)` | `> 1` |
`3x – x^2 – 12 + 4x` | `> 1` |
`-x^2 + 7x – 13` | `> 0` |
`Delta = 7^2 – 4 ⋅ 1 ⋅ 13 = -3 < 0`
`=>\ text(No Solutions)`
`text(If)\ \ x – 4 < 0, x < 4`
`-(x – 4) (3 – x)` | `> 1` |
`x^2 – 7x + 12` | `> 1` |
`x^2 – 7x + 11` | `> 0` |
`x` | `= (7 +- sqrt(7^2 – 4 ⋅ 1 ⋅ 11))/2` |
`= (7 +- sqrt 5)/2` |
`text(Combining solutions)`
`(x < (7 – sqrt 5)/2 ∪ x > (7 + sqrt 5)/2) nn x < 4`
`x ∈ (– oo, (7 – sqrt 5)/2)`
Solve `3/(|\ x - 3\ |) < 3`. (3 marks)
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`x < 2\ ∪\ x > 4`
`text(Solution 1)`
`3/(|\ x – 3\ |) < 3`
`|\ x – 3\ |` | `> 1` |
`(x^2 – 6x + 9)` | `> 1^2` |
`x^2 – 6x + 8` | `> 0` |
`(x – 4)(x – 2)` | `> 0` |
`:. {x: \ x < 2\ ∪\ x > 4}`
`text(Solution 2)`
`|\ x – 3\ | > 1`
`text(If)\ \ (x – 3)` | `> 0,\ text(i.e.)\ x >3` |
`x – 3` | `> 1` |
`x` | `> 4` |
`=> x > 4\ (text(satisfies both))`
`text(If)\ \ (x – 3)` | `< 0,\ text(i.e.)\ x <3` |
`−(x – 3)` | `> 1` |
`−x + 3` | `> 1` |
`x` | `< 2` |
`=> x < 2\ (text(satisfies both))`
`:. {x: \ x < 2\ ∪\ x > 4}`
Indicate the region on the number plane satisfied by `y ≥ |\ x + 1\ |.` (2 marks)
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`text(See Worked Solution)`
Find the values of `x` for which `|\ x − 3\ | ≤ 1`. (2 marks)
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`2 ≤ x ≤ 4`
Find the values of `x` for which `|\ x + 1\ |<= 5`. (2 marks)
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`-6 <= x <= 4`
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i. |
ii. `text(Solving for)\ \ |\ 2x – 1\ | <= |\ x – 3\ |`
`text(Graph shows the statement is TRUE)`
`text(between the points of intersection.)`
`=>\ text(Intersection occurs when)`
`(2x – 1)` | `= (x – 3)\ \ \ text(or)\ \ \ ` | `-(2x – 1)` | `= x – 3` |
`x` | `= -2` | `-2x + 1` | `= x – 3` |
`-3x` | `= -4` | ||
`x` | `= 4/3` |
`:.\ text(Solution is)\ \ {x: -2 <= x <= 4/3}`
Which inequality has the same solution as `|\ x + 2\ | + |\ x- 3\ | = 5`?
(A) `5/(3 - x) >= 1`
(B) `1/(x - 3)\ - 1/(x + 2) <= 0`
(C) `x^2 - x - 6 <= 0`
(D) `|\ 2x - 1\ | >= 5`
`C`
`text(In)\ A\ text(and)\ B, \ x ≠ 3\ text(but when)\ x=3,`
`|\ 3 + 2\ | + |\ 3 – 3\ | = 5\ \ text(is correct.)`
`:.\ text(Not)\ \ A\ \ text(or)\ \ B.`
`text(Consider)\ D`
`x -> oo\ text(satisfies)\ |\ 2x – 1\ | >= 5,\ \ text(but)`
`text(obviously not)\ |\ x + 2\ | + |\ x – 3\ | = 5.`
`text(Consider)\ C`
`x^2 – x – 6` | `<= 0` |
`(x – 3)(x + 2)` | `<= 0` |
`text(True when)\ \ \ -2 <= x <= 3.`
`text(In this range,)`
`(x + 2) >= 0\ \ text(and)\ \ (x – 3)<= 0`
`:.\ text(We can write)`
`|\ x + 2\ | + |\ x – 3\ |` | `= (x + 2)\ – (x – 3)` |
`= x + 2 – x + 3` | |
`= 5` |
`:. C\ text(has the same solution)`
`=> C\ text(is correct.)`
The diagram shows the graphs `y = |\ x\ |\ - 2` and `y = 4- x^2`.
Write down inequalities that together describe the shaded region. (2 marks)
`text(Inequalities are)`
`y <= 4\ – x^2`
`y >= |\ x\ |\ – 2`
`text(Inequalities are)`
`y <= 4 – x^2`
`y >= |\ x\ |\ – 2`
Solve `|\ 3x -1\ | < 2` (2 marks)
` -1/3 < x < 1 `
`|\ 3x -1\ | < 2`
`3x -1` | `<2` | `\ \ \ \ \-(3x -1)` | `< 2` |
`3x` | `<3` | `-3x + 1` | `< 2` |
`x` | `< 1` | `3x` | `> -1` |
`x` | `> -1/3` |
`:. -1/3 < x < 1`