### Solving equations and inequalities: Linear inequalities in one unknown

### Solving a linear inequality via equations

You can also solve a linear inequality by

- first replacing the inequality sign by an equal sign,
- then solving this equation, and
- finally, determining the sign of the inequality for point to the left and to the right of the solution of the equation.

Determine the exact solution of the inequality \[x + 5 \lt -3x -2\] via equations.

\(x \lt -{{7}\over{4}}\)

We follow the following roadmap:

- Get started with the corresponding equation \[x + 5 = -3x -2\]
- Solve this equation:

- Get the terms with \(x\) on the left-hand side of the equation (by adding \(3x\) on both sides):

\(x + 5 +3x = -3x -2 +3x\), which simplifies to \(4x +5 = -2\). - Then move the terms without \(x\) to the right (by adding \(-5\) both sides):

\(4x +5 - 5 = -2 - 5\), which simplifies to \(4x = -7\).- Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(4\)); this gives \(x = \;\frac{-7}{4}\).

- So, the solution of the equation is \(x = {-{{7}\over{4}}}\).

- Get the terms with \(x\) on the left-hand side of the equation (by adding \(3x\) on both sides):

- Find out whether the solutions are on the number line to the left or to the right of \(-{{7}\over{4}}\).
- First calculate the left- and right-hand sides of the inequality \(x + 5 \lt -3x -2\) when you substitute a value of \(x\) less than \(-{{7}\over{4}}\). For example, when you fill in \(x=-10\), then you get \(-5 \lt 28\) and this is a true statement. Any other value of \(x\) less than \(-{{7}\over{4}}\) may be used too, and you still get a true statement.
- Then calculate the left- and right-hand sides of the inequality \(x + 5 \lt -3x -2\) when you substitute a value of \(x\) greater than \(-{{7}\over{4}}\). For example, when you fill in \(x=10\), then you get \(15 \lt -32\) and this is a false statement. Any other value of \(x\) greater than \(-{{7}\over{4}}\) may be used too, and you still get a false statement.
- From these two numeric examples follows that solutions \(x\) of \(x + 5 \lt -3x -2\) must satisfy \(x \lt -{{7}\over{4}}\).

The points where the inequality holds are shown in green in the number line below. An open circle around \(x=-{{7}\over{4}}\) indicates that we are dealing with an inequality of the type \(\lt\) or \(\gt\), where in this case the point itself is not a solution. A closed circle indicates an inequality of the type \(\le\) or \(\ge\), and then the point marked on the number line is element of the solution set.