Exercises: Inequalities

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.

Solve the following linear inequality:

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

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

Solve the following system of inequalities in one variable:

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

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

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

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

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 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 in two variables whose solution corresponds to the shaded region in the following figure.

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?

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?

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

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

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 $10$).

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

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 unknown):

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

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.

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?

Solve the following inequality combining modulus and quadratic polynomials:

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

A balanced diet requires that the daily intake of two foods, $A$ and $B$, provides at least $60$ 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.