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 factored: $(x - 4)(x + 2) < 0$.

Solve the following incomplete quadratic inequality and represent its solution on the real 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 sign of the numerator and the denominator: $\frac{x-3}{x+1} \leq 0$.

The sum of twice 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 so that its length is $4$ feet greater than its width $x$. If there is enough fertilizer to cover a maximum of $45 \text{ ft}^2$, set up an inequality and determine the possible values that the width $x$ can take.

A small craft business has fixed monthly costs of $\$300$ and producing each leather bracelet costs $\$2$. If it sells each bracelet for $\$5$, what is the minimum number of bracelets it must sell per month 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 the terms to one side and operate to obtain a single quotient before studying the signs.

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

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

A delivery van consumes $8$ gallons of fuel per $100 \text{ mi}$ on the highway and $12$ gallons per $100 \text{ mi}$ in the city. If on a trip of $500 \text{ mi}$ in total it has consumed less than $45$ gallons, what is the maximum number of miles it could have traveled 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 in which 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 what particularity its solution set has:

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

$2x - 1 > 0$

A water rocket is launched vertically upwards in a school experiment. Its height $h$ in feet 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 $17 \text{ ft}$ in height.

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

The monthly profits of an electric scooter factory, expressed in thousands of dollars, are 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 unequal 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, it is desired that its perimeter does not exceed $36 \text{ cm}$. Set up a system of inequalities and find the possible values of $x$.

Study and solve the following system formed by a linear inequality and a quadratic inequality:

$3x - 5 \leq x + 1$

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

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