Question

Solve the inequality for real $x$: $37 - (3x + 5) \geqslant 9x - 8(x - 3)$

Answer 1

Hint: To solve linear inequality in one variable we will simplify it in similar way of linear equation and we will take variable terms to the one side and constants to the other side and then we will simplify it for value of variable(x) but we should keep in mind the notation of inequality which decides the interval of solution.

Complete step by step answer:
Since given that the equation $37 - (3x + 5) \geqslant 9x - 8(x - 3)$ and then we need to find the value of the unknown variable $x$ for the real number system, so we will make use of the basic mathematical operations to simplify further.
By the multiplication operation, we get $37 - 3x - 5 \geqslant 9x - 8x + 24$
now Turing the variables on the left-hand side and also the numbers on the right-hand side we get $37 - 3x - 5 \geqslant 9x - 8x + 24 \Rightarrow - 3x - 9x + 8x \geqslant 24 + 5 - 37$ while changing the values on the equals to, the sign of the values or the numbers will change.
Hence by the addition and subtraction operation, we have, $ - 12x + 8x \geqslant 24 - 32 \Rightarrow - 4x \geqslant - 8$
Thus, by division, we get $ - 4x \geqslant - 8 \Rightarrow 4x \leqslant 8 \Rightarrow x \leqslant \dfrac{8}{4} \Rightarrow x \leqslant 2$ where $ - x \geqslant - y \Rightarrow x \leqslant y$
Hence, we have the values as $x \leqslant 2$
Since the unknown variable $x$ is given as real, which means Natural numbers are $\{ 1,2,3,...\infty \} $ and whole numbers defined as $\{ 0,1,2,3,...\infty \} $ and integers defined as $\{ - \infty ,..., - 3, - 2, - 1,0,1,2,3,...\infty \} $ and the rational and irrational also, hence real numbers means all should be contained in them, they $x \leqslant 2$ can be expressed as a real form of $x \in ( - \infty ,2]$ where the number $2$ is at most.

Note:
Generally students make mistakes while multiplying the inequality by minus 1 as they forget to change the sign of inequality. And sometimes students forget to check that boundary points are included or excluded.