Inequalities
At a taco stand, each taco costs $\$20$ and Ximena has $\$150$. She doesn't ask "How many tacos do I buy?" but "How many tacos can I buy?" That is the difference: an inequality does not have a single answer, it has a set of answers. At the end of the page we will solve Ximena's case.
Intervals and what it means to solve an inequality
For example, $-3 \le x < 5$ is written $[-3, 5)$: $-3$ is included and $5$ is not.
Write as an interval and represent on the number line the values of $x$ that satisfy the inequality $-3 \le x < 5$. You can do it on paper and send a photo with your…
Write as an interval and represent on the number line the values of $x$ that satisfy the inequality $-3 \le x < 5$.
You can do it on paper and send a photo with your answer.
First degree
They are solved just like an equation, with one extra rule: when you multiply or divide both sides by a negative number, the inequality reverses direction.
Why? We know that $2 < 5$. If we multiply by $-1$ we get $-2$ and $-5$, and now $-2 > -5$: when we switch to negatives, the order is reversed.
With denominators, first multiply by the LCM, which is positive, so the direction does not change.
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 brackets: $\frac{3(x - 2)}{4} - \frac{2x + 1}{6} \le \frac{x - 3}{2} + 1$
Solve the linear inequality with fractions and brackets:
$\frac{3(x - 2)}{4} - \frac{2x + 1}{6} \le \frac{x - 3}{2} + 1$
From everyday language to inequalities
To set up a problem you have to translate the phrases correctly. This table helps you:
| In words | Symbol |
|---|---|
| less than, fewer than | $<$ |
| at most, no more than, a maximum of | $\le$ |
| more than, greater than | $>$ |
| at least, a minimum of, no less than | $\ge$ |
Now we close Ximena's case: $20x \le 150$ gives $x \le 7.5$. Since tacos come in whole numbers, she can buy 7 at most.
Translate into algebraic language and solve: "Three times a number, decreased by $4$ units, is at most $11$".
Translate into algebraic language and solve: "Three times a number, decreased by $4$ units, is at most $11$".
To pass a subject, a student must have a mean mark of at least $5$ marks across the three term exams. If in the first two exams they scored $3.5$ and $4.2$, what mark…
To pass a subject, a student must have a mean mark of at least $5$ marks across the three term exams. If in the first two exams they scored $3.5$ and $4.2$, what mark must they get in the third exam to pass? (The maximum mark is $10$).
Second degree
It is solved in three steps: move everything to one side to compare with $0$, find the roots (by factoring or with the quadratic formula), and study the sign interval by interval.
Why does it work? An expression like $(x-3)(x+2)$ can only change sign at its roots. Between two consecutive roots the sign does not change, so it is enough to test one value in each interval.
If the coefficient of $x^2$ is negative, you can multiply by $-1$ reversing the direction, or else study the signs of the parabola that opens downward.
Solve the quadratic inequality $x^2 - 16 < 0$. Express the solution as an interval.
Solve the quadratic inequality $x^2 - 16 < 0$. Express the solution as an interval.
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 upward. The height $h$ (in meters) reached by the rocket as a function of time $t$ (in seconds) is given by the expression…
A toy rocket is launched vertically upward. The height $h$ (in meters) reached by the rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = 30t - 5t^2$. During which time interval will the rocket be at a height strictly greater than $25$ meters?
When the inequality depends on a parameter
A parabola $y = x^2 - mx + 9$ opens upward. It will be positive for all $x$ when it does not touch the $x$-axis: that is, when it has no real roots, or in other words, when the discriminant $b^2 - 4ac$ is negative.
Determine the values of the parameter $m$ so that the quadratic inequality $x^2 - mx + 9 > 0$ holds for any real number $x$.
Determine the values of the parameter $m$ so that the quadratic inequality $x^2 - mx + 9 > 0$ holds for any real number $x$.
With unknowns in denominators and higher degree
With an $x$ in the denominator you do not multiply by the denominator, because we don't know whether it is positive or negative. Instead:
-
Move everything to one side so that one side equals $0$.
-
Combine into a single fraction.
-
Study the sign of the numerator and the denominator interval by interval.
The roots of the denominator never belong to the solution, because there the expression does not exist.
For a higher-degree polynomial, factor it first and make the same sign table with all the roots.
Solve the following rational inequality, analyzing the sign in the corresponding intervals: $\frac{2x - 5}{x + 3} \le 0$
Solve the following rational inequality, analyzing the sign in the corresponding intervals:
$\frac{2x - 5}{x + 3} \le 0$
Solve the higher-degree polynomial inequality by first factorising the polynomial: $x^3 - 4x^2 - x + 4 \le 0$
Solve the higher-degree polynomial inequality by first factorising the polynomial:
$x^3 - 4x^2 - x + 4 \le 0$
Solve the rational inequality, paying attention to simplifying correctly before studying the signs (remember that you cannot eliminate denominators containing the…
Solve the rational inequality, paying attention to simplifying correctly before studying the signs (remember that you cannot eliminate denominators containing the unknown):
$\frac{x + 1}{x - 2} > \frac{1}{2}$
Systems of inequalities
Solve the following system of inequalities in one variable: $\begin{cases} 2x - 3 > 1 \\ x + 4 \le 7 \end{cases}$
Solve the following system of inequalities in one variable:
$\begin{cases} 2x - 3 > 1 \\ x + 4 \le 7 \end{cases}$
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:
$\begin{cases} \frac{x + 1}{2} - \frac{x - 2}{3} > x - 1 \\ 3x - 4(x + 1) \ge -6 \end{cases}$
The base of a rectangle measures $5$ cm more than its height. If the perimeter of the rectangle must be greater than $30$ cm but its area cannot exceed $150$ cm$^2$,…
The base of a rectangle measures $5$ cm more than its height. If the perimeter of the rectangle must be greater than $30$ cm but its area cannot exceed $150$ cm$^2$, between what values does the measure of the height lie?
Inequalities with two unknowns
An inequality like $x + y \le 4$ has as its solution a region of the plane (a half-plane), not a single point. The line $x + y = 4$ is the boundary: it is drawn solid if the inequality is $\le$ or $\ge$ (the points on the line are included) and dashed if it is $<$ or $>$.
To check a point: substitute its coordinates. If the inequality holds, the point is in the region. To know which side to shade: test an easy point (almost always the origin) that is not on the line.
Check whether the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality in two variables $3x - 2y \le 4$.
Check whether the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality in two variables $3x - 2y \le 4$.
A farmer wants to bottle his oil production. Knowing that he has $2$-liter bottles and $5$-liter bottles available, and that he has strictly less than $100$ liters of…
A farmer wants to bottle his oil production. Knowing that he has $2$-liter bottles and $5$-liter bottles available, and that he has strictly less than $100$ liters of oil, write a linear inequality in two variables that models this situation. If he has already filled $10$ $5$-liter bottles, what is the maximum number of $2$-liter bottles he can fill?
Write the system of inequalities in two variables whose solution corresponds to the shaded region in the following figure.
Write the system of inequalities in two variables whose solution corresponds to the shaded region in the following figure.
Feasible region and linear programming
A system of inequalities with two unknowns is solved by drawing each half-plane; the solution is the area where all of them overlap. That area is called the feasible region, and each inequality in the system is a constraint.
To read a system from a region: for each side of the region, find the equation of the line that contains it and decide the direction of the inequality by testing a point inside the region.
Often it is not enough to know what is possible: we want the best possible, for example the greatest profit. That quantity is called the objective function (here, a linear expression in $x$ and $y$).
A small company manufactures two types of wireless headphones: model Alfa and model Beta. Model Alfa requires $3$ hours of assembly and $1$ hour of calibration, while…
A small company manufactures two types of wireless headphones: model Alfa and model Beta. Model Alfa requires $3$ hours of assembly and $1$ hour of calibration, while model Beta 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 $20$ Alfa headphones and $25$ Beta headphones in a week.
Observe the following bounded feasible region (a triangle) and determine the system of three linear inequalities in two unknowns that defines it.
Observe the following bounded feasible region (a triangle) and determine the system of three linear inequalities in two unknowns that defines it.
A balanced diet requires that the daily intake of two foods, $A$ and $B$, provides at least $60$ g of protein and does not exceed $80$ g of fat. Each $100$ g portion of…
A balanced diet requires that the daily intake of two foods, $A$ and $B$, provides at least $60$ g of protein and does not exceed $80$ g of fat. Each $100$ g portion of food $A$ contains $15$ g of protein and $10$ g of fat. Each $100$ g portion of food $B$ contains $10$ g of protein and $20$ g of fat. If we let $x$ and $y$ be the number of $100$ g portions of food $A$ and $B$ consumed respectively, set up the system of inequalities and find out whether it is possible to consume exactly $3$ portions of $A$ and $2$ portions of $B$ while satisfying the constraints.
Absolute value
Solve the absolute value inequality $|x - 3| \le 5$ and express its solution using an interval.
Solve the absolute value inequality $|x - 3| \le 5$ and express its solution using an interval.
Solve the following inequality combining modulus and quadratic polynomials: $|x^2 - 5x| < 6$
Solve the following inequality combining modulus and quadratic polynomials:
$|x^2 - 5x| < 6$
Summary
Back to the beginning: Ximena, with $\$150$ and tacos at $\$20$, solved $20x \le 150$ and got $x \le 7.5$. She can buy at most $7$ tacos.