Exercises: Inequalities with one unknown

Solve the following linear inequality: $3(x - 2) + 4 \leq 5x - 8$. Express the solution in interval notation.

Solve the inequality with denominators: $\frac{x}{2} - \frac{x-1}{3} > 1$.

Find the values of $x$ that satisfy the quadratic inequality, which is already given in factorised form: $(x - 4)(x + 2) < 0$.

Solve the following incomplete quadratic inequality and represent its solution on the number line: $2x^2 - 18 \geq 0$.

Find the solution interval of the quadratic inequality: $x^2 - 5x + 6 \leq 0$.

Solve the following system of inequalities with one unknown and express the solution as a single interval:

$2x - 3 < 5$

$3x + 1 \geq -2$

Solve the simple rational inequality by studying the signs of the numerator and denominator: $\frac{x-3}{x+1} \leq 0$.

The sum of double a natural number and $5$ is strictly less than $17$. Set up the corresponding inequality and find all the natural numbers that satisfy it.

Solve the following inequality, paying attention to the negative signs in front of the fractions: $\frac{2x - 1}{4} - \frac{3x + 2}{6} \geq \frac{x - 5}{3} - 1$.

A rectangular garden is to be built such that its length is $4$ meters greater than its width $x$. If there is enough fertilizer to cover a maximum of $45 \text{ m}^2$, set up an inequality and determine the possible values that the width $x$ can take.

A small craft business has monthly fixed costs of $6\,000$ pesos and producing each leather bracelet costs $40$ pesos. If it sells each bracelet for $100$ pesos, how many bracelets must it sell per month as a minimum to start making a profit? Set up and solve the inequality.

Solve the rational inequality: $\frac{2x + 1}{x - 2} > 3$. Remember to move all terms to one side and simplify to obtain a single fraction before studying the signs.

Solve the higher-degree polynomial inequality by first taking out a common factor: $x^3 - 4x^2 - 5x \leq 0$.

Solve the inequality with modulus and express its solution as a union of intervals: $|2x - 5| \geq 7$.

A delivery van consumes $8$ liters of fuel every $100 \text{ km}$ on the highway and $12$ liters every $100 \text{ km}$ in the city. If on a trip of $500 \text{ km}$ in total it has consumed less than $45$ liters, what is the maximum number of kilometers it could have driven in the city?

Determine for which values of the parameter $k$ the quadratic equation $x^2 - kx + 9 = 0$ has no real solutions. Set up the corresponding inequality using the discriminant and solve it.

Solve the rational inequality where the numerator and the denominator are quadratic polynomials: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.

Solve the following system of inequalities and explain the special feature of its solution set:

$x^2 - 4x + 4 \leq 0$

$2x - 1 > 0$

A water rocket is launched vertically upward in a school experiment. Its height $h$ in meters as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine algebraically during which time interval the rocket will be above a height of $17 \text{ m}$.

Solve the modulus inequality that includes fractional expressions: $\left| \frac{3x - 1}{2} \right| < 4$.

The monthly profit of an electric scooter factory, expressed in thousands of pesos, is given by the function $B(x) = -x^3 + 10x^2 - 16x$, where $x$ represents the number of hundreds of scooters manufactured. How many scooters must be manufactured so that the company does not make a loss (that is, so that $B(x) \geq 0$)?

In an isosceles triangle, the equal sides measure $x \text{ cm}$ and the third side measures $10 \text{ cm}$. For the triangle to be constructed, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Furthermore, its perimetre must not exceed $36 \text{ cm}$. Set up a system of inequalities and find the possible values of $x$.

Study and solve the following system consisting of a linear inequality and a quadratic inequality:

$3x - 5 \leq x + 1$

$x^2 - 7x + 10 > 0$

Solve the following rational inequality, simplifying the algebraic additions required to express it as a single quotient before studying the signs: $\frac{1}{x} + \frac{1}{x+2} \geq \frac{3}{4}$.