Question

Solve the inequality $\left\{ {3\left( {2x - 5} \right) - 7} \right\} \geqslant 9\left( {x - 5} \right)$.

Answer 1

Hint: Here, we will proceed by simplifying the given inequality which contains only one variable (i.e., x) in such a way that only x occurs on the LHS of the inequality. This will help to obtain the range of values of x for which the given inequality is satisfied.

Complete Step-by-Step solution:
Given equality is $\left\{ {3\left( {2x - 5} \right) - 7} \right\} \geqslant 9\left( {x - 5} \right){\text{ }} \to (1)$
$
   \Rightarrow \left\{ {6x - 15 - 7} \right\} \geqslant 9x - 45 \\
   \Rightarrow 6x - 22 \geqslant 9x - 45 \\
 $
Taking 6x from the LHS to the RHS and taking 45 from the RHS to the LHS, we get
$
   \Rightarrow 45 - 22 \geqslant 9x - 6x \\
   \Rightarrow 23 \geqslant 3x \\
 $
By reversing the above inequality, we get
$
   \Rightarrow x \leqslant \dfrac{{23}}{3} \\
   \Rightarrow x \leqslant 7.667 \\
 $
So, only those values of x are accepted which are less than 7.667.
Therefore, the values of x which satisfies the given inequality ranges from negative infinity up to 7.667.

Note: In this particular problem, we have used the basic concepts like whenever a term is shifting from the RHS or the LHS of an inequality to the LHS or the RHS respectively of that inequality, the sign of that term changes (i.e., positive term becomes negative term and negative term becomes positive term).