Inequalities
Worked Examples
Intro to Inequalities
We say that $a< b$ if $b-a$ is a positive real number, or equivalently, if $a$ is to the left of $b$ on the number line. (We read $a< b$ as "$a$ is less than $b$.")The set of all points $x$ on the number line that satisfy the inequality $x< a$ are those to the left of $a$ on the number line. We call this set of points, the solution set of the inequality $x< a$.
We sketch this set as follows:
where the open circle $\quad \circ \quad$ indicates that the point $a$ is not included in the set.
We say that $a \leq b$ if either $a< b$ or $a=b$. (We read $a \leq b$ as "$a$ is less than or equal to $b$.") The set of all points $x$ on the number line that satisfy the inequality $x\leq a$ are those points to the left of $a$, together with $a$.
We sketch them as follows:
where the closed circle $\quad \bullet \quad$ indicates that the point $a$ is included in the set.
Similarly, the solution to $x >a$ is all points to the right of $a$.
We sketch it as follows:
The solution to $x\geq a$ is all the points to the right of $a$, together with $a$.
We sketch it as follows:
We can also combine inequalities, such as $a\leq x < b$, by solving each inequality that appears separately, then taking the common solution.
Example.
The double inequality $1< x \leq 3$ can be solved by finding the points, which are solution to $1 < x$ and $x \geq 3$ at the same time. This gives the set of all points to the right of 1 and the left of 3, together with the point 3.We sketch it as follows:
In many ways, solving inequalities works the same way as solving equalities. However, there are some important differences. Here are the rules for manipulating inequalities.
(We write them down only for "$ < $", however they apply to $\leq, >, \geq$ as well.)
| (1) | You can add or subtract a constant to both sides of an inequality:
If $a < b$, then $a \pm c < b \pm c$. E.g. $1 < 2 \; \Rightarrow \; 1+1 < 2+1$, or $\; 2 < 3$. |
| (2) | You can multiply both sides of an inequality by a positive constant:
If $a < b$ and $c > 0$, then $ca < cb$. E.g. $1<2 \; \Rightarrow \; 2 \cdot 1 < 2 \cdot 2$, or $\; 2 < 4$. |
| (3) | If you multiply both sides of an inequality by a negative constant, then you must reverse the inequality:
If $a < b$ and $c < 0$, then $ca > cb$. E.g. $1 < 2 \; \Rightarrow \; (-1) \cdot 1 > (-1) \cdot 2$, or $\; -1 > -2$. |
| (4) | If $0 < a < b$, then $\frac{1}{a} > \frac{1}{b}$.
E.g. $2<3 \; \Rightarrow \; \frac{1}{2} > \frac{1}{3}$ |
We can use these rules to simplify inequalities that we are trying to solve, like we do when solving equalities.
Example. Solve $3x-5 < 13$ and sketch the solution on the number line.
| $ 3x-5<13$ | ||
| $3x<18$ | (Details)add $5$ to both sides | |
| $x<6\; \;$ | (Details)multiply both sides by $\frac{1}{3}$ or divide both sides by $3$ |
