Question

How do you solve the inequalities $3x - 5 > 10$ or $3 - 2x < - 13$?

Answer 1

Hint: In this question, we need to find the solution for the inequality given. So, let us consider the given expressions of inequality first and then start solving. To get the solution we have gathered all the x terms at one side and all other constant terms at another side. To do this we use some mathematical operations such as addition, subtraction, multiplication and try to simplify the given inequality. Make sure that the variable x is at L.H.S. and then find the solution for the problem. Then we get the required solution for the given inequalities.

Complete step-by-step solution:
For solving the question, let us consider the given expression of inequalities
$3x - 5 > 10$ or $3 - 2x < - 13$
We are asked to find the solution for the above inequalities.
For solving the given problem, we have to bring all the terms containing x on one side and the constant terms on the other side.
Consider the first inequality given by $3x - 5 > 10$ …… (1)
Now add 5 to both sides of the equation (1), we get,
$ \Rightarrow 3x - 5 + 5 > 10 + 5$
Combining the like terms $ - 5 + 5 = 0$
Combining the like terms $10 + 5 = 15$
Hence we have,
$ \Rightarrow 3x + 0 > 15$
$ \Rightarrow 3x > 15$
Dividing by 3 on both sides we get,
$ \Rightarrow \dfrac{{3x}}{3} > \dfrac{{15}}{3}$
$ \Rightarrow x > 5$
Now consider the second inequality given by $3 - 2x < - 13$ …… (2)
Now subtract 3 on both sides of the equation (2), we get,
$ \Rightarrow 3 - 2x - 3 > - 13 - 3$
Rearranging the terms we get,
$ \Rightarrow - 2x + 3 - 3 > - 13 - 3$
Combining the like terms $3 - 3 = 0$
Combining the like terms $ - 13 - 3 = - 16$
Hence we have,
$ \Rightarrow - 2x + 0 > - 16$
$ \Rightarrow - 2x > - 16$
Dividing by -2 on both sides we get,
$ \Rightarrow \dfrac{{ - 2x}}{{ - 2}} > \dfrac{{ - 16}}{{ - 2}}$
$ \Rightarrow x > 8$
Hence by solving the inequality $3x - 5 > 10$ we get the solution as, $x > 5$ and for $3 - 2x < - 13$ we get the solution as, $x > 8$.

Note: Remember that when transferring any variable or number to the other side, the signs of the same will be changed to the opposite sign.
When both sides of the equation are added or subtracted by a positive number the inequality sign remains the same and when both sides of the equation are multiplied or divided by a negative number then the inequality gets reversed.
In solving such problems, we must be careful about which number must be added or subtracted, so that the problem gets simplified and it becomes easier to find out the solution.