Question

How do you solve the inequality $d + 6 \leqslant 4d - 9$ or $3d - 1 < 2d + 4$?

Answer 1

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.