audio read-through Solving Linear Inequalities in One Variable

In Precalculus, it's essential that you can easily and efficiently solve sentences like:

$\displaystyle\frac{3x}{2}-1 \ge \frac 15 - 7x$
$\displaystyle x^2 \ge 3$

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.

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Solving Linear Inequalities

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:

Let's look at an example to understand this situation.

In the sketch below, ‘$\,a\lt b\,$’ is true, because $\,a\,$ lies to the left of $\,b\,.$

Multiplying both sides of the inequality by $\,-1\,$ sends $\,a\,$ to its opposite ($\,-a\,$) , and sends $\,b\,$ to its opposite ($\,-b\,$) . Now, the opposite of $\,a\,$ is to the right of the opposite of $\,b\,.$ That is, $\, -a\gt -b\,.$

multiplying by -1 sends a number to its opposite

So, multiplying two numbers by $\,-1\,$ changes their left/right positioning relative to each other!

For example, $\,1 \lt 2\,$ (one is to the left of two), but $\,-1 \gt -2\,$ (negative one is to the right of negative two). That's why you need to change the inequality symbol!


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

$$\cssId{s50}{\frac{3x}{2}-1} \cssId{s51}{\ \ge\ \frac 15 - 7x}$$
Original inequality
$$\cssId{s53}{10\bigl(\frac{3x}{2}-1\bigr)} \cssId{s54}{\ \ge\ 10(\frac 15 - 7x)}$$
To clear fractions, multiply both sides by $\,10\,$ (which is the least common multiple of $\,2\,$ and $\,5\,$).
$$\cssId{s56}{15x - 10} \cssId{s57}{\ \ge\ 2 - 70x}$$
After multiplying out, all fractions are gone.
$$\cssId{s59}{-10} \cssId{s60}{\ \ge\ 2 -85x}$$
Subtract $\,15\,x$ from both sides.
$$\cssId{s62}{-12} \cssId{s63}{\ \ge\ -85x}$$
Subtract $\,2\,$ from both sides.
$$\cssId{s65}{\frac{-12}{-85}} \cssId{s66}{\ \le\ \frac{-85x}{-85}}$$
Divide both sides by $\,-85\,$; change direction of inequality symbol.
$$\cssId{s68}{\frac{12}{85}} \cssId{s69}{\ \le\ x}$$
$$\cssId{s71}{x} \cssId{s72}{\ \ge\ \frac{12}{85}}$$
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 an optional 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!

Concept Practice

For graphical insight into the solution set, a graph is optionally 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 to toggle the graph.