MAT 101: Solving Linear Inequalities

Linear Inequalities:

Solving linear inequalities is as easy as solving linear equations. You only need to know the following principles.

For a given inequality,

1. Adding the same number to each side of the inequality does not change the symbol $<$.
2. Subtracting the same number from each side of the inequality does not change the symbol $<$.
3. Multiplying or dividing the inequality by the same positive number does not change the symbol $<$.
4. Multiplying or dividing the inequality by the same negative number reverses the symbol $<$ from, say, $<$ to $>$.

The principle 4 can be easily understood with a simple example, say everybody would agree that $1<2$. If one multiplies the inequality by $-1$, it is true that $-1>-2$, i.e. the symbol $<$ reverses.

Example. Solve the inequality
$$3x-5<6-2x.$$

Solution. Adding 5 to each side of the inequality results
$$3x<11-2x.\ \ \ \ \ \mbox{(1)}$$
Adding $2x$ to each side of (1) results
$$5x<11.\ \ \ \ \ \mbox{(2)}$$
Dividing the inequality (2) by the same positive number 5 results
$$x<\frac{11}{5}.$$
There are various ways to write the solution. Normally $x<\frac{11}{5}$ would suffice but it can be written more formally as the solution set
$$\left\{x|\ x<\frac{11}{5}\right\}.$$
In interval notation, the solution can be written as
$$\left(-\infty,\frac{11}{5}\right).$$
Graphically it can be represented as

Example. Solve the inequality
$$13-7x\geq 10x-4.$$

Solution. Subtracting 13 from each side of the inequality results
$$-7x\geq 10x-17.\ \ \ \ \ \mbox{(3)}$$
Subtracting $10x$ from each side of (3) results
$$-17x\geq -17.\ \ \ \ \ \mbox{(4)}$$
Dividing each side of (4) by the same negative number $-17$ results
$$x\leq 1.$$

Note that the symbol $\geq$ has been reversed to $\leq$. The solution set can be written as $\{x|\ x\leq 1\}$ or $(-\infty,1]$ in interval notation. Graphically it can be represented as

Compound Inequalities:

A compound inequality is two inequalities joined by a conjunction AND or OR.

Example. Solve $-3<2x+5\leq 7$.

Solution. Notice that the compund inequality is formed by
$$-3<2x+5\ \mbox{and}\ 2x+5\leq 7.$$
Subtract 5 from each side of the given inequality.
$$-8<2x\leq 2.\ \ \ \ \ \mbox{(5)}$$
Devide (5) by 2.
$$-4<x\leq 1.$$
Hence the solution set is $\{x|\ -4<x\leq 1\}$ or $(-4,1]$ in interval notation. The graph of the solution set is given by

Example. Solve $2x-5\leq -7$ or $2x-5>1$.

Solution. Add 5 to eqch side of the two inequalities:
$$2x\leq -2\ \mbox{or}\ 2x>6.\ \ \ \ \ \mbox{(6)}$$
Divide each side of the two inequalities (6) by 2:
$$x\leq -1\ \mbox{or}\ x>3.$$
The solution set is $\{x|\ x\leq -1\ \mbox{or}\ x>3\}$ or $(-\infty,-1]\cup(3,\infty)$ in interval notation. The graph of the solution set is given by