Question

How do you solve the inequality \[ - \dfrac{3}{4}x < 20\]?

Answer 1

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 ‘x’ 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 solve this for ‘x’.

Complete step-by-step solution:
Given, \[ - \dfrac{3}{4}x < 20\]
Now we need to solve this for ‘x’.
Now if we multiply by 4 on both side we know that the direction of the inequality changes,
\[
   - \dfrac{3}{4} \times 4x < 20 \times 4 \\
   - 3x < 80 \\
 \]
Divide by negative 3 on both sides and we know that the direction of the inequality changes,
\[x > - \dfrac{{80}}{3}\]
Thus the solution of \[ - \dfrac{3}{4}x < 20\] is \[x > - \dfrac{{80}}{3}\]. In interval form \[\left( { - \dfrac{{80}}{3},\infty } \right)\].

Note: 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.