Question

How do you solve the inequality $3-2x>7$?

Answer 1

To solve this linear inequality in one variable, we have to take the variable terms to one side of the inequality, and the constant terms to the other side. Inequalities do not provide a fixed value as the solution, it gives a range. All the values in this range hold the inequality. To solve the inequality we should know some of the properties of the inequality as follows, given that$a> b$. We can state the following from this.
$a+k> b+k,k\in$Real numbers
$ak> bk,k\in$Positive real numbers
$ak< bk,k\in$Negative real numbers

Complete step-by-step solution:
We are given the inequality, $3-2x>7$.
Subtracting 7 from both sides of the above inequality, we get
$\Rightarrow 3-2x-7>7-7$
$\Rightarrow -2x-4>0$
Adding $2x$ to both sides of the above inequality, we get

& \Rightarrow -2x-4+2x>0+2x \\\ & \Rightarrow -4>2x \\\ \end{aligned}$$ By multiplying or dividing an inequality by a positive quantity, the inequality sign does not change. Dividing both sides of the above inequality by 2, we get $$\Rightarrow -\dfrac{4}{2}>\dfrac{2x}{2}$$ Canceling out 2 as a common factor of numerator and denominator for the RHS of inequality, we get $$\Rightarrow -2>x$$ **Hence the solution of the given inequality is $$x <-2$$.** **Note:** We can also use a different property to solve the given inequality, as follows $$3-2x>7$$ Subtracting 3 from both sides of the inequality, we get $$\begin{aligned} & \Rightarrow 3-2x-3>7-3 \\\ & \Rightarrow -2x>4 \\\ \end{aligned}$$ By multiplying or dividing an inequality by a negative quantity, the inequality sign changes. Dividing both sides of the above inequality by $$-2$$, we get $$\begin{aligned} & \Rightarrow \dfrac{-2x}{-2}<\dfrac{4}{-2} \\\ & \Rightarrow x<-2 \\\ \end{aligned}$$ We are getting the same range from both of the methods. We can check whether the solution is correct or not by substituting any value in the range we got. Let’s substitute $$x=-3$$ in the inequality, the LHS of the inequality becomes $$\Rightarrow 3-2(-3)=9$$ And the RHS is $$7$$. From this as, $$LHS>RHS$$ the solution is correct.