Answer

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**Hint:**An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘d’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality\[( \leqslant , > )\]. We have simple inequality and we can solve them easily.

**Complete step by step solution:**

Now take,

\[d + 6 \leqslant 4d - 9\]

We know that the direction of inequality doesn't change if we add or subtract a positive number on both sides.

\[\Rightarrow d + 6 + 9 \leqslant 4d - 9 + 9\]

\[\Rightarrow d + 6 + 9 \leqslant 4d\]

\[\Rightarrow d + 15 \leqslant 4d\]

We subtract ‘d’ on both sides,

\[\Rightarrow d - d + 15 \leqslant 4d - d\]

\[\Rightarrow 15 \leqslant 3d\]

Swapping left and right hand side then we have,

\[\Rightarrow 3d \geqslant 15\]

Divide by 3 on both sides we have,

\[\Rightarrow d \geqslant \dfrac{{15}}{3}\]

\[ \Rightarrow d \geqslant 5\]

That is the solution of \[d + 6 \leqslant 4d - 9\] is \[d \geqslant 5\]. In interval form \[[5,\infty )\].

Now take \[3d - 1 < 2d + 4\] and following the same steps as above,

Add 1 on both sides of the equation,

\[\Rightarrow 3d - 1 + 1 < 2d + 4 + 1\]

\[\Rightarrow 3d < 2d + 5\]

Now subtract ‘2d’ on both sides of the inequality,

\[\Rightarrow 3d - 2d < 2d - 2d + 5\]

\[ \Rightarrow d < 5\]

That is the solution of \[3d - 1 < 2d + 4\] is \[d < 5\]. In interval form \[( - \infty ,5)\].

(if we have \[ \leqslant \] and \[ \geqslant \] we will have closed interval if not we will have open interval)

**Note:**For the inequality \[d + 6 \leqslant 4d - 9\] if we take ‘d’ value in \[[5,\infty )\]and put it in \[d + 6 \leqslant 4d - 9\]. It satisfies

Put \[d = 5\] in \[d + 6 \leqslant 4d - 9\],

\[5 + 6 \leqslant 4(5) - 9\]

\[11 \leqslant 20 - 9\]

\[11 \leqslant 11\]

It is correct because 11 is equal to 11. We check for the second inequality also.

We know that \[a \ne b\] says that ‘a’ is not equal to ‘b’. \[a > b\] means that ‘a’ is less than ‘b’. \[a < b\] means that ‘a’ is greater than ‘b’. These two are known as strict inequality. \[a \geqslant b\] means that ‘a’ is less than or equal to ‘b’. \[a \leqslant b\] means that ‘a’ is greater than or equal to ‘b’.

**The direction of inequality do not change in these cases:**

i) Add or subtract a number from both sides.

ii) Multiply or divide both sides by a positive number.

iii) Simplify a side.

**The direction of the inequality change in these cases:**

i) Multiply or divide both sides by a negative number.

ii) Swapping left and right hand sides.

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