Question

How do you solve the inequality \[3x+8>2\]?

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
\[ak< bk,k\in \] Negative real numbers

Complete step by step solution:
We are asked to solve the inequality \[3x+8>2\]. 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 or subtracting a number to both sides of an inequality the sign of inequality does not change. Subtracting 8 from both sides of the inequality, we get
\[\begin{align}
  & \Rightarrow 3x+8>2 \\
 & \Rightarrow 3x>2-8 \\
 & \Rightarrow 3x>-6 \\
\end{align}\]
By multiplying a positive quantity to both sides of an inequality the inequality sign does not change. dividing both sides of the above inequality by 3, we get
\[\begin{align}
  & \Rightarrow \dfrac{3x}{3}>\dfrac{-6}{3} \\
 & \Rightarrow x>-2 \\
\end{align}\]
Hence, the solution range for given inequality is \[x\in \left( -2,\infty \right)\].

Note: We can check if the solution is correct or not by substituting any value in the range we got. Let’s substitute \[x=0\] in the given inequality, we get
\[\Rightarrow 3(0)+8>2\]
Simplifying the above expression, we get
\[\Rightarrow 8>2\]
This is a true statement. Hence, the solution range we got is correct.