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Solve `3-x/5<8` if `x` is a negative number. (2 marks)
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Solve `(4x)/5+1> -1` and graph the solution on a number line. (3 marks)
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`x> -5/2`
| `(4x)/5+1` | `> -1` | |
| `(4x)/5` | `> -2` | |
| `4x` | `> -10` | |
| `x` | `> -10/4` | |
| `x` | `> -5/2` |
In this inequality `n` is a whole number.
`8/n < 5/8`
What is the smallest possible value for `n` to make this inequality true? (2 marks)
`4/45 = 3/40 + 1/x`
Find the value of `x?` (2 marks)
`2/13 < 3/x` where `x` is a positive whole number.
What is the highest possible value for `x`? (2 marks)
Solve `3/(x+2) < 4`. (3 marks)
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`text(Solution 1)`
`3/(x + 2) < 4`
`text(Multiply b.s. by)\ \ (x + 2)^2`
| `3(x + 2)` | `< 4(x + 2)^2` |
| `3x + 6` | `< 4 (x^2 + 4x + 4)` |
| `3x + 6` | `< 4x^2 + 16x + 16` |
| `4x^2 + 13x + 10` | `> 0` |
| `(4x + 5)(x + 2)` | `> 0` |
`text(LHS)\ = 0\ \ text(when)\ \ x = -5/4\ \ text(or)\ \ -2`
`text(From graph)`
`x < -2\ \ text(or)\ \ x > -5/4`
`text(Alternate Solution)`
`text(If)\ \ x + 2 > 0\ \ \ \ text{(i.e.}\ \ x > –2 text{)}`
| `3` | `< 4(x + 2)` |
| `3` | `< 4x + 8` |
| `4x` | `> -5` |
| `x` | `> -5/4` |
`text(If)\ \ x + 2 < 0\ \ \ \ text{(i.e.}\ x < –2 text{)}`
| `3` | `> 4 (x + 2)` |
| `3` | `> 4x + 8` |
| `4x` | `< -5` |
| `x` | `< -5/4` |
| `:. x` | `< –2\ \ \ \ text{(satisfies both)}` |
`:.\ x < –2\ \ text(or)\ \ x > –5/4`
Solve `(4-x)/x <1`. (3 marks)
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`text(Solution 1)`
`(4-x)/x < 1`
| `text(If)\ x<0,\ \ \ \ \ 4-x` | `> x` |
| `2x` | `< 4` |
| `x` | `<2` |
`=> x<0\ \ \ text{(satisfies both)}`
| `text(If)\ x>0,\ \ \ \ \ 4-x` | `<x` |
| `2x` | `>4` |
| `x` | `>2` |
`=> x>2\ \ \ text{(satisfies both)}`
`:.\ x < 0\ \ text(or)\ \ x > 2`
`text(Solution 2)`
`text(Multiply both sides by)\ \ x^2`
| `x(4-x)` | `< x^2` |
| `4x-x^2` | `< x^2` |
| `2x^2-4x` | `>0` |
| `2x(x-2)` | `>0` |
`text(From graph)`
`x<0\ \ text(or)\ \ x >2`
