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

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Last updated date: 04th Mar 2024
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IVSAT 2024
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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