Inequalities
At an amusement park, the sign at the roller coaster says: You must be at least 48 inches tall to ride. Which heights get you on? Not one number, but a whole range of them. An equation asks "which value?" An inequality asks "which values?" In this page you will learn to describe those ranges, solve every common type of inequality, and even draw the answer when there are two variables.
Intervals and the number line
Closed circle means the endpoint is included ($\le$, $\ge$). Open circle means it is not included ($<$, $>$).
| Inequality | Interval | Graph |
|---|---|---|
| $x > a$ | $(a, \infty)$ | open circle at $a$, shade right |
| $x \ge a$ | $[a, \infty)$ | closed circle at $a$, shade right |
| $x < a$ | $(-\infty, a)$ | open circle at $a$, shade left |
| $x \le a$ | $(-\infty, a]$ | closed circle at $a$, shade left |
| $a \le x < b$ | $[a, b)$ | closed at $a$, open at $b$, shade between |
Use $\cup$ ("union") to join two separate pieces, such as $(-\infty, -2) \cup (3, \infty)$. Use $\cap$ ("intersection") for the values two sets have in common.
Write in interval notation and represent on the real line the values of $x$ that satisfy the inequality $-3 \le x < 5$.
Write in interval notation and represent on the real line the values of $x$ that satisfy the inequality $-3 \le x < 5$.
Linear inequalities
A linear inequality is solved just like a linear equation, with one new rule.
Inequalities with fractions
Clear the denominators first by multiplying every term by the least common denominator. The LCD is positive, so the symbol does not flip.
Solve the following linear inequality: $4x - 5(x - 2) \ge 3x + 2$
Solve the following linear inequality:
$4x - 5(x - 2) \ge 3x + 2$
Solve the linear inequality with denominators: $\frac{x}{2} - \frac{x - 1}{3} > 1$
Solve the linear inequality with denominators:
$\frac{x}{2} - \frac{x - 1}{3} > 1$
Solve the linear inequality with fractions and parentheses: $\frac{3(x - 2)}{4} - \frac{2x + 1}{6} \le \frac{x - 3}{2} + 1$
Solve the linear inequality with fractions and parentheses:
$\frac{3(x - 2)}{4} - \frac{2x + 1}{6} \le \frac{x - 3}{2} + 1$
Compound inequalities and systems
A system of inequalities in one variable (or a compound inequality joined by and) asks for the values that satisfy all the conditions at once. Solve each inequality separately, then take the intersection.
Why? "And" means a number is accepted only if it passes every test. Where two graphs overlap, both tests pass.
A double inequality like $-3 \le 2x + 1 < 9$ is a system in disguise. Do the same operation on all three parts: $-4 \le 2x < 8$, so $-2 \le x < 4$.
Solve the following system of inequalities with one unknown: $\begin{cases} 2x - 3 > 1 \\ x + 4 \le 7 \end{cases}$
Solve the following system of inequalities with one unknown:
$\begin{cases} 2x - 3 > 1 \\ x + 4 \le 7 \end{cases}$
Find the solution to the following system of inequalities: $\begin{cases} \frac{x + 1}{2} - \frac{x - 2}{3} > x - 1 \\ 3x - 4(x + 1) \ge -6 \end{cases}$
Find the solution to the following system of inequalities:
$\begin{cases} \frac{x + 1}{2} - \frac{x - 2}{3} > x - 1 \\ 3x - 4(x + 1) \ge -6 \end{cases}$
To pass a class, a student must have an average score of at least $5$ points across the three exams of the term. If they got a $3,5$ and a $4,2$ on the first two exams,…
To pass a class, a student must have an average score of at least $5$ points across the three exams of the term. If they got a $3,5$ and a $4,2$ on the first two exams, what score must they get on the third exam to pass? (The maximum score is $10$).
Quadratic inequalities
Think of the parabola $y = ax^2 + bx + c$. Solving $ax^2 + bx + c < 0$ means "where is the parabola below the x-axis?" Solving $> 0$ means "where is it above?" The x-intercepts (zeros) are where the sign can change.
Solve the quadratic inequality $x^2 - 16 < 0$. Express the solution in interval notation.
Solve the quadratic inequality $x^2 - 16 < 0$. Express the solution in interval notation.
Solve the quadratic inequality $-2x^2 + 5x + 3 \ge 0$. Express the solution as an interval.
Solve the quadratic inequality $-2x^2 + 5x + 3 \ge 0$. Express the solution as an interval.
A toy rocket is launched vertically upwards. The height $h$ (in feet) reached by the rocket as a function of time $t$ (in seconds) is given by the expression…
A toy rocket is launched vertically upwards. The height $h$ (in feet) reached by the rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = 30t - 5t^2$. During what time interval will the rocket be at a height strictly greater than $25$ feet?
Always true, never true, and parameters
Sometimes the parabola never reaches the x-axis. Then the quadratic has no real zeros, and its sign never changes.
| Parabola opens up ($a > 0$) | Discriminant $b^2 - 4ac$ | What happens |
|---|---|---|
| no real zeros | negative | $ax^2 + bx + c > 0$ for all real $x$ |
| one zero (touches the axis) | zero | $> 0$ for all $x$ except the zero |
| two zeros | positive | negative between them, positive outside |
For example, $x^2 + 4 > 0$ is true for every real number, and $x^2 + 4 < 0$ has no solution. Likewise $(x - 2)^2 \le 0$ is true only at $x = 2$.
Move the slider $k$ below and watch where the parabola touches the x-axis.
Determine the values of the parameter $m$ so that the quadratic inequality $x^2 - mx + 9 > 0$ is satisfied for any real number $x$.
Determine the values of the parameter $m$ so that the quadratic inequality $x^2 - mx + 9 > 0$ is satisfied for any real number $x$.
Word problems with one variable
Most mistakes in word problems come from the translation, so learn the key phrases.
| Words | Symbol |
|---|---|
| less than, fewer than, below | $<$ |
| more than, greater than, exceeds, over | $>$ |
| at most, no more than, maximum, does not exceed | $\le$ |
| at least, no less than, minimum | $\ge$ |
Remember that "decreased by $4$" means subtract $4$. "Twice a number plus $7$ is at least $19$" becomes $2n + 7 \ge 19$, so $n \ge 6$.
Translate into algebraic language and solve: "Three times a number, decreased by $4$, is at most $11$."
Translate into algebraic language and solve: "Three times a number, decreased by $4$, is at most $11$."
A farmer wants to bottle their oil production. Knowing that they have $2$-gallon and $5$-gallon bottles available, and that they have strictly less than $100$ gallons of…
A farmer wants to bottle their oil production. Knowing that they have $2$-gallon and $5$-gallon bottles available, and that they have strictly less than $100$ gallons of oil, write a linear inequality with two unknowns that models this situation. If they have already filled $10$ bottles of $5$ gallons, what is the maximum number of $2$-gallon bottles they can fill?
The base of a rectangle measures $5$ in. more than its height. If the perimeter of the rectangle must be greater than $30$ in. but its area cannot exceed $50$ in$^2$,…
The base of a rectangle measures $5$ in. more than its height. If the perimeter of the rectangle must be greater than $30$ in. but its area cannot exceed $50$ in$^2$, between what values does the height measure lie?
Absolute value inequalities
The expression $|x - a|$ is the distance between $x$ and $a$ on the number line. So $|x - 2| \le 4$ means "$x$ is at most $4$ units away from $2$."
Always isolate the absolute value first. If $k < 0$, no distance is below a negative, so $|u| < k$ has no solution, and $|u| > k$ is true for every $x$.
Solve the absolute value inequality $|x - 3| \le 5$ and express its solution as an interval.
Solve the absolute value inequality $|x - 3| \le 5$ and express its solution as an interval.
Solve the following inequality that combines absolute value and quadratic polynomials: $|x^2 - 5x| < 6$
Solve the following inequality that combines absolute value and quadratic polynomials:
$|x^2 - 5x| < 6$
Rational inequalities
A rational inequality has the unknown in a denominator. The tempting move is to multiply both sides by the denominator, but you can't: you don't know whether it is positive or negative, so you don't know whether to flip the symbol.
Solve the following rational inequality, studying the sign in the corresponding intervals: $\frac{2x - 5}{x + 3} \le 0$
Solve the following rational inequality, studying the sign in the corresponding intervals:
$\frac{2x - 5}{x + 3} \le 0$
Solve the rational inequality, paying attention to calculating correctly before studying the signs (remember that you cannot eliminate denominators containing the…
Solve the rational inequality, paying attention to calculating correctly before studying the signs (remember that you cannot eliminate denominators containing the unknown):
$\frac{x + 1}{x - 2} > \frac{1}{2}$
Higher-degree polynomial inequalities
The sign-chart method works for any polynomial that you can factor. Factor completely, mark every zero, and test each interval. A simple (single) zero makes the sign change.
If the polynomial does not factor by grouping, look for an integer zero by trying small numbers such as $\pm 1, \pm 2$, then divide it out.
Solve the higher-degree polynomial inequality by factoring the polynomial first: $x^3 - 4x^2 - x + 4 \le 0$
Solve the higher-degree polynomial inequality by factoring the polynomial first:
$x^3 - 4x^2 - x + 4 \le 0$
Inequalities in two variables
The solution of an inequality like $2x - y < 4$ is a region of the coordinate plane, containing every point $(x, y)$ that makes it true.
To check whether a point is a solution, substitute its coordinates and see whether the inequality is true.
Check if the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality with two unknowns $3x - 2y \le 4$.
Check if the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality with two unknowns $3x - 2y \le 4$.
Systems of inequalities and graphs
A system of inequalities in two variables is solved by the overlap of the shaded half-planes. The overlap is called the feasible region. It is bounded (a polygon) when enough inequalities close it in.
To go the other way, from a graph to a system, treat each boundary line separately.
Write the system of inequalities with two unknowns whose solution corresponds to the shaded region in the following figure.
Write the system of inequalities with two unknowns whose solution corresponds to the shaded region in the following figure.
Look at the following bounded feasible region (a triangle) and determine the system of three linear inequalities with two unknowns that defines it.
Look at the following bounded feasible region (a triangle) and determine the system of three linear inequalities with two unknowns that defines it.
Modeling with systems of inequalities
When resources are limited (time, money, ingredients), each limit becomes an inequality. Together with $x \ge 0$ and $y \ge 0$ (you cannot make a negative number of items), they define the feasible region. A plan is possible only if its point lies inside the region, which means it satisfies every inequality.
A small company manufactures two types of drones: the Alpha model and the Beta model. The Alpha model requires $3$ hours of assembly and $1$ hour of calibration, while…
A small company manufactures two types of drones: the Alpha model and the Beta model. The Alpha model requires $3$ hours of assembly and $1$ hour of calibration, while the Beta model requires $2$ hours of assembly and $2$ hours of calibration. The company has a maximum of $120$ hours for assembly and $80$ hours for calibration per week. Set up the system of inequalities that defines the feasible region and state whether it would be possible to manufacture $20$ Alpha drones and $25$ Beta drones in a week.
A balanced diet requires that the daily intake of two foods, $A$ and $B$, provides at least $60$ oz of protein and does not exceed $80$ oz of fat. Each $100$ oz portion…
A balanced diet requires that the daily intake of two foods, $A$ and $B$, provides at least $60$ oz of protein and does not exceed $80$ oz of fat. Each $100$ oz portion of food $A$ contains $15$ oz of protein and $10$ oz of fat. Each $100$ oz portion of food $B$ contains $10$ oz of protein and $20$ oz of fat. If we call $x$ and $y$ the number of $100$ oz portions of foods $A$ and $B$ consumed respectively, set up the system of inequalities and find out if it is possible to consume exactly $3$ portions of $A$ and $2$ portions of $B$ while fulfilling the restrictions.
Back to the start: the roller coaster rule is $h \ge 48$, so every height in $[48, \infty)$ rides, and $47.9$ in. does not.