## Linear Inequalities

Linear inequalities are often easy to solve. We just need to isolate the variable on one side of the inequality sign.

- Recall that we can
- add (or subtract) the same quantity (+ or $-$) from both sides of the inequality, or
- multiply (or divide) both sides of the inequality by a positive quantity, and the inequality will still hold.

- If we multiply or divide both sides of an inequality by a negative quantity, the direction of the inequality will be reversed (See Section 1.4).

** Never multiply or divide both sides of an inequality by a quantity whose sign is unknown!**

- For part (b) of the above example, note that

\[-4<\dfrac{7-5x}{2}\leq 1 \] means

\[-4<\dfrac{7-5x}{2}\qquad\text{and}\qquad\dfrac{7-5x}{2}\leq1.\] In fact, we have to solve two inequalities.

## Nonlinear Inequalities

To solve nonlinear inequalities, we follow these steps:

- Move all terms to one side of the inequality sign and express the inequality in the form

\[P>0,\qquad\text{or}\qquad P<0\] or

\[P\geq0,\qquad\text{or}\qquad P\leq 0 \] where $P$ is an expression in the variable (usually $x$) [to indicate its dependence on $x$, we may write it as $P(x)$ if we wish] - Factor $P$

\[

P=Q_{1}Q_{2}\cdots Q_{n}

\] where $Q_{1},…,Q_{n}$ are expressions in $x$. - Determine the zeros of each factor of $P$ (find the values for which each $Q_{i}$ is zero). These values divide the real line into intervals.
- Make a sign table. Determine the sign of each factor in each interval.
- Determine the sign of $P$ in each interval using the sign table. Recall that a product (or a quotient) that involves an even number of negative factors is positive and one that involves an odd number of negative factors is negative. If the inequality sign is $\geq$ or $\leq$, pay attention to the endpoints of the intervals and check if they satisfy the inequality.

- For fractions; that is, when

\[

P=\frac{Q_{1}}{Q_{2}}

\] we follow the same steps. - To determine the sign of each factor, we can choose an arbitrary number (called a
**test value**) in that interval and find the sign of the factor. Alternatively, you can use the following facts:

- If $ax^{2}+bx+c$ does not have any real roots, then its sign is always the same as the sign of $a$

[because in this case the graph of $y=ax^{2}+bx+c$ is completely above or completely below the $x$-axis, and never intersects it]

- If $ax^{2}+bx+c$ has two real roots $r_{1}$ and $r_{2}$, then when $x$ is between $r_{1}$ and $r_{2}$ , its sign is the opposite of the sign of $a$ and outside that interval, its sign is the same as the sign of $a$.

[Because when $ax^{2}+bx+c$ has two real roots, then

\[ax^{2}+bx+c=a(x-r_{1})(x-r_{2}) \]