Definition
Linear inequality
A linear inequality is an inequality that may be put in the following forms:
ax+b\leq cx+d
ax+b\geq cx+d
Where x is a variable and a,b,c,d are real numbers.
The following inequality is an example of a linear inequality:
2x-3\leq x+1
The following are also linear inequalities:
- ax+b\lt cx+d
- ax+b\gt cx+d
Solving linear inequalities with simple operations
In order to solve a linear inequality, isolate the variable in the inequality using simple operations.
Note that when multiplying or dividing by negative numbers, we must reverse the inequality sign.
Consider the following linear inequality:
-2x-3\leq 4
To isolate the variable x, begin by adding 3 to both sides of the inequality:
-2x-3+3\leq 4+3\\-2x\leq 7
To isolate the variable x, divide by -2 and reverse the inequality sign since we are dividing by a negative number.
\dfrac{-2x}{-2}\geq \dfrac{7}{-2}\\x\geq -\dfrac{7}{2}
The solution set to the inequality is the interval \left[-\dfrac{7}{2},\infty \right)
A linear inequality with variables on both sides can be solved by isolating the variables on one side.
Consider the following linear inequality:
3x-1\lt 2x+3
The linear inequality is solved by isolating the variable x, which can be done as follows:
3x-1-2x\lt 2x+3-2x\\x-1\lt 3\\x-1+1\lt 3+1\\x\lt 4
Therefore, the solution set to the linear inequality is the interval \left(-\infty,4\right).
Solving linear inequalities with graphs
Linear inequalities can be solved graphically.
Consider the following linear inequality:
3x-1\lt 2x+3
To solve the linear inequality graphically, graph the lines y=3x-1 and y=2x+3. The x -values of the points on the line y=3x-1 strictly below the line y=2x+3 correspond to the solutions of the linear inequality.
The line y=3x-1 lies strictly beneath line y=2x+3 and to the left of the intersection point \left(4{,}11\right). Therefore, the solution set to the linear inequality is the interval \left(-\infty,4\right).
Consider the following linear inequality:
2x+3\geq x-1
The following graphic contains the graph of the equations y=2x+3 and y=x-1.
The line y=2x+3 lies either on or above the line y=x-1 and to the right of the point of the intersection \left(-4,-5\right). Therefore, the set of solutions to the linear inequality is the interval \left[-4,\infty\right).
Systems of linear inequalities
System of linear inequalities
A system of linear inequalities is a collection of linear inequalities.
The following is a system of linear inequalities:
\begin{cases} 2x-3\leq 1 \cr \cr x+1\geq 0 \cr \cr 3x+1\lt 2x+8 \end{cases}
Solving a system of linear inequalities requires solving each linear inequality in the system and only allowing for solutions which solves all linear inequalities in the system.
Consider the following system of linear inequalities:
\begin{cases} 2x-1\lt 1 \cr \cr x\geq 0 \end{cases}
The linear inequality 2x-1\lt 1 is solved as follows:
2x-1\lt 1\\2x\lt 2\\x\lt 1
Thus the solution set to the first linear inequality is the inequality \left(-\infty,1\right).
In the second inequality, the variable x is already isolated and the solution set to the second linear inequality is the interval \left[0,\infty\right).
The set of x -values common to both intervals is the interval \left[0{,}1\right), which can also be seen in the following graphic:
Some system of linear inequalities have no solutions.
Consider the following set of linear inequalities:
\begin{cases} x\geq 0 \cr \cr x-1\lt-4 \end{cases}
The solution set of the first inequality is the interval \left[0,\infty\right), and the solution set of the second inequality is the interval \left(-\infty,-3\right). There is no overlap between these intervals and there are therefore no solutions to this system of inequalities.
