Exercises: Inequalities

Write in interval notation and represent on the real line the values of $x$ that satisfy the inequality $-3 \le x < 5$.

Solve the following linear inequality:

$4x - 5(x - 2) \ge 3x + 2$

Solve the quadratic inequality $x^2 - 16 < 0$. Express the solution in interval notation.

Solve the following system of inequalities with one unknown:

$\begin{cases} 2x - 3 > 1 \\ x + 4 \le 7 \end{cases}$

Translate into algebraic language and solve: "Three times a number, decreased by $4$, is at most $11$."

Check if the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality with two unknowns $3x - 2y \le 4$.

Solve the absolute value inequality $|x - 3| \le 5$ and express its solution as an interval.

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 quadratic inequality $-2x^2 + 5x + 3 \ge 0$. Express the solution as an interval.

Find the solution to the following system of inequalities:

$\begin{cases} \frac{x + 1}{2} - \frac{x - 2}{3} > x - 1 \\ 3x - 4(x + 1) \ge -6 \end{cases}$

Write the system of inequalities with two unknowns whose solution corresponds to the shaded region in the following figure.

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$^2$, between what values does the height measure lie?

Solve the following rational inequality, studying the sign in the corresponding intervals:

$\frac{2x - 5}{x + 3} \le 0$

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 $10$).

Determine the values of the parameter $m$ so that the quadratic inequality $x^2 - mx + 9 > 0$ is satisfied for any real number $x$.

Solve the higher-degree polynomial inequality by factoring the polynomial first:

$x^3 - 4x^2 - x + 4 \le 0$

Solve the rational inequality, paying attention to calculating correctly before studying the signs (remember that you cannot eliminate denominators containing the unknown):

$\frac{x + 1}{x - 2} > \frac{1}{2}$

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.

Look at the following bounded feasible region (a triangle) and determine the system of three linear inequalities with two unknowns that defines it.

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?

Solve the following inequality that combines absolute value and quadratic polynomials:

$|x^2 - 5x| < 6$

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.