Inequality#
Inequalities play a significant role in various fields such as economics, physics, and optimization problems, where they are used to represent constraints and relationships between variables that are not necessarily equal.
An inequality is a mathematical statement that expresses a relationship between two values. Inequalities are used to compare quantities that are not necessarily equal.
Inequalities are denoted using the following symbols:
\(>\) Greater than
\(<\) Less than
\(\ge\) Greater than or equal to
\(\leq\) Less than or equal to
For example
\(x > 5\) means “\(x\) is greater than 5.”
\(y < 10\) means “\(y\) is less than 10.”
\(z \ge 3\) means “\(z\) is greater than or equal to 3.”
\(w \le 7\) means “\(w\) is less than or equal to 7.”
Inequalities can also involve variables and constants, similar to equations. Solving inequalities involves determining the possible range of values for the variable that satisfy the inequality. In other words, we are find out what are all the possible values of the unknown variable in order to have the inequality true.
Solving Linear Inequalities#
When solving linear inequalities (inequalities with a variable raised to the power of 1), the same rules for manipulating equations apply, with one key difference: when you multiply or divide both sides of the inequality by a negative number, the inequality sign reverses. Here’s a general process.
Perform operations on both sides of the inequality to isolate the variable.
Pay attention to sign changes when multiplying or dividing by negative numbers.
Express the solution as an interval or a set of values.
For example, to solve \(2x - 3 < 5\), you would add 3 to both sides to isolate \(2x\) on the LHS. Then divide the inequality by 2 in order to find the solution to be \(x < 4\). The solution states that
The inequality \(2x - 3 < 5\) is true, when the variable \(x\) is lesser than 4. The solution can also be expressed as a set of \(\{3, 2, 1, 0, -1, -2, \dots\}\).
Examples#
\(\begin{align}\text{a)} \ 5x-5&<3x+9 \\ 5x-3x&<9+5 \\ 2x&<14 \ \|:2 \\ &x<7\end{align}\)
\(\begin{align}\text{b)} \ 7 & \le 3x-4 \\ -3x & \le -11 \ \|:(-3) \\ x & \ge \frac{11}{3}\end{align}\)
The division by a negative number can be avoided, if wanted, as in the example below:
\(\begin{align} 7 & \le 3x-4 \\ 11 & \le 3x \\ 3x & \ge 11 \ \|:3 \\ x & \ge \frac{11}{3}\end{align}\)
\(\begin{align}\text{c)} \ -\frac{x}{4}&>-3 \ \| \cdot(-4) \\ x&<12\end{align}\)