by Dr. Carol JVF Burns (website creator)
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In Precalculus, it's essential that you can easily and efficiently solve sentences like:

‘$\,\frac{3x}{2}-1 \ge \frac 15 - 7x\,$’         and         ‘$\,x^2 \ge 3\,$’

This lesson gives you practice with the first type of sentence, which is a linear inequality in one variable. The next lesson gives you practice with the second type of sentence, which is a nonlinear inequality in one variable.

This page gives an in-a-nutshell review of essential concepts and skills for solving linear inequalities in one variable.
If you're needing more, a thorough review is also outlined.

All links to lessons open in a new window, so you don't lose this page.


The sentence ‘$\,\frac{3x}{2}-1 \ge \frac 15 - 7x\,$’ is an example of a linear inequality in one variable.

Only extremely basic tools are needed to solve linear inequalities in one variable:

So, solution of the sample sentence could look something like this:

$$ \eqalign{ \cssId{s50}{\frac{3x}{2}-1} \ &\cssId{s51}{\ge\ \frac 15 - 7x} \qquad\qquad\qquad&\cssId{s52}{\text{original inequality}}\cr\cr \cssId{s53}{10\bigl(\frac{3x}{2}-1\bigr)} \ &\cssId{s54}{\ge\ 10(\frac 15 - 7x)} &\cssId{s55}{\text{to clear fractions, multiply both sides by 10, which is the least common multiple of 2 and 5}}\cr\cr \cssId{s56}{15x - 10} \ &\cssId{s57}{\ge\ 2 - 70x} &\cssId{s58}{\text{after multiplying out, all fractions are gone}}\cr\cr \cssId{s59}{-10} \ &\cssId{s60}{\ge\ 2 -85x} &\cssId{s61}{\text{subtract $\,15\,x$ from both sides}}\cr\cr \cssId{s62}{-12} \ &\cssId{s63}{\ge\ -85x} &\cssId{s64}{\text{subtract $\,2\,$ from both sides}}\cr\cr \cssId{s65}{\frac{-12}{-85}} \ &\cssId{s66}{\le\ \frac{-85x}{-85}} &\cssId{s67}{\text{divide both sides by $\,-85\,$, change direction of inequality symbol}}\cr\cr \cssId{s68}{\frac{12}{85}} \ &\cssId{s69}{\le\ x} &\cssId{s70}{\text{simplify}}\cr\cr \cssId{s71}{x} \ &\cssId{s72}{\ge\ \frac{12}{85}} &\cssId{s73}{\text{write in more conventional form, with the variable on the left}} } $$

By the way, head up to and type in     3x/2 -1 >= 1/5 - 7x    (you can cut-and-paste). Voila!

For a much more thorough review of solving linear inequalities, do this:

Now, practice solving some fairly complicated linear inequalities by click-click-clicking below.
All steps in the solution process are shown.

Each problem is accompanied by a graph, which offers additional insights into the solution set.
If the information contained in the graph isn't meaningful to you, first study the next section,
Solving Nonlinear Inequalities in One Variable (graphical concepts), and then come back!

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Solving Nonlinear Inequalities in One Variable

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

Solve the given inequality.
Write the result in the most conventional way.

For graphical insight into the solution set, a graph is displayed.
For example, the inequality $\,-6 - 3x \ge 4\,$ is optionally accompanied by
the graph of $\,y = -6 - 3x\,$ (the left side of the inequality, dashed green)
and the graph of $\,y = 4\,$ (the right side of the inequality, solid purple).
In this example, you are finding the values of $\,x\,$ where
the green graph lies on or above the purple graph.
Click the “show/hide graph” button if you prefer not to see the graph.

1 2 3 4 5 6 7 8

(MAX is 8; there are 8 different problem types.)