Question

How do you solve the inequality \[\dfrac{1}{3}x-4<-10\]?

Answer 1

Hint: 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 a solution; it gives a range. All the values in this range hold the inequality. To solve an 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
Complete step by step answer:
We are asked to solve the inequality \[\dfrac{1}{3}x-4<-10\]. We will use the properties of inequalities to solve this question. The properties we will use are 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
By adding a number to both sides of an inequality the sign of inequality does not change. Adding 4 to both sides of the inequality, we get
\[\begin{align}
  & \Rightarrow \dfrac{1}{3}x-4+4<-10+4 \\
 & \Rightarrow \dfrac{1}{3}x<-6 \\
\end{align}\]
By multiplying a positive quantity to both sides of an inequality the inequality sign does not change. Multiplying both sides of the above inequality by 3, we get
\[\begin{align}
  & \Rightarrow \left( \dfrac{1}{3}x \right)3<-6\times 3 \\
 & \Rightarrow x<-18 \\
\end{align}\]

Hence, the solution range for given inequality is \[x\in \left( -\infty ,-18 \right)\].

Note: We can check if the solution is correct or not by substituting any value in the range we got. Let’s substitute \[x=-30\] in the given inequality, we get
\[\dfrac{1}{3}\left( -30 \right)-4<-10\]
Cancelling out common factors from numerator and denominator from the left sides of inequality, we get
\[\begin{align}
  & \Rightarrow -10-4<-10 \\
 & \Rightarrow -14<-10 \\
\end{align}\]
This is a true statement. Hence, the solution range we got is correct.