First- and second-degree inequalities
When a problem says "at least", "must not exceed" or "from how many onwards", the mathematical translation is not an equation but an inequality: a statement that can have infinitely many solutions, a whole interval of the real line. This section covers first-degree inequalities (where the sign matters when multiplying or dividing) and second-degree inequalities (where the sign is studied interval by interval).
First-degree inequalities
Solve the following first-degree inequality and represent the solution on the real number line using an interval: $5 - 2x \le 11$ Draw a real number line in your…
Solve the following first-degree inequality and represent the solution on the real number line using an interval:
$5 - 2x \le 11$
Draw a real number line in your notebook with a suitable scale, mark the endpoint of the interval and indicate whether it is included or not. Then write the solution using interval notation.
Check whether the values $x = 0$, $x = 2$ and $x = 5$ are solutions of the quadratic inequality: $x^2 - 5x + 4 \le 0$
Check whether the values $x = 0$, $x = 2$ and $x = 5$ are solutions of the quadratic inequality:
$x^2 - 5x + 4 \le 0$
Second-degree inequalities: roots and sign table
A group of friends wants to organise a graduation party. The venue rental costs a fixed $150$ €, and the catering costs $12$ € per attendee. If they have a maximum…
A group of friends wants to organise a graduation party. The venue rental costs a fixed $150$ €, and the catering costs $12$ € per attendee. If they have a maximum budget of $600$ €, set up an inequality and calculate the maximum number of people who can attend the party.
A company that manufactures phone cases has determined that its monthly profits, in hundreds of euros, depend on the number of units produced $x$ (in thousands)…
A company that manufactures phone cases has determined that its monthly profits, in hundreds of euros, depend on the number of units produced $x$ (in thousands) according to the function $B(x) = -2x^2 + 14x - 12$.
Between which production values does the company make a profit (that is, $B(x) > 0$)? Solve the inequality and express the solution as an interval, in thousands of units.
Special cases: negative discriminant and repeated root
If the discriminant is negative, the parabola does not cross the x-axis: it is always positive or always negative, depending on the sign of $a$. If the discriminant is zero, there is a repeated root and the sign is the same on both sides.
Determine the solution set of the following quadratic inequality by analyzing the sign of the factors: $x^2 - 9 > 0$ Express the result as intervals.
Determine the solution set of the following quadratic inequality by analyzing the sign of the factors:
$x^2 - 9 > 0$
Express the result as intervals.
Find the solution of the following system of linear inequalities with one unknown and express it as an interval: $\begin{cases} x + 3 > 1 \\ 2x - 4 \le 2 \end{cases}$…
Find the solution of the following system of linear inequalities with one unknown and express it as an interval:
$\begin{cases} x + 3 > 1 \\ 2x - 4 \le 2 \end{cases}$
You can use this number line to represent the solutions of each inequality and their intersection:
Systems of inequalities in one unknown
A system of inequalities is solved in the same way as a system of equations: each inequality separately, and at the end the intersection of the solutions. Mixing a linear and a quadratic inequality does not change the method; it just requires care with the sign in each interval.
A group of 1st-year Bachillerato students is organizing a graduation party. They have calculated that the fixed costs of renting the venue and the DJ are $450$ €. In…
A group of 1st-year Bachillerato students is organizing a graduation party. They have calculated that the fixed costs of renting the venue and the DJ are $450$ €. In addition, the catering costs $12$ € per attendee. If they have raised a total of $1500$ € through raffle ticket sales and sponsorships, and they want to have at least $210$ € left over after the event for an end-of-year trip, write and solve an inequality that determines the maximum number of people, $x$, that they can invite.
In the graph you can see the money raised and the costs as a function of the number of attendees.
A consulting services firm estimates that its monthly profits, expressed in thousands of euros, are given by the rational expression $B(x) = \frac{x^2 - 9}{x - 5}$,…
A consulting services firm estimates that its monthly profits, expressed in thousands of euros, are given by the rational expression $B(x) = \frac{x^2 - 9}{x - 5}$, where $x$ represents the number of projects managed simultaneously. For operational reasons, the firm can only manage between 0 and 10 projects. Determine the range of projects for which the firm achieves positive profits that are less than or equal to 4 thousand euros, that is, solve the system:
$0 < \frac{x^2 - 9}{x - 5} \leq 4$
Real-life problems
Find the set of real numbers that simultaneously satisfy the following system of inequalities: $$\begin{cases} x^2 - 5x + 4 \leq 0 \\ 2x + 1 > 5 \end{cases}$$
Find the set of real numbers that simultaneously satisfy the following system of inequalities:
$$\begin{cases} x^2 - 5x + 4 \leq 0 \\ 2x + 1 > 5 \end{cases}$$An electronic components manufacturer has found that the unit production cost $C$ depends on the room temperature $T$ (in ºC) according to the expression…
An electronic components manufacturer has found that the unit production cost $C$ depends on the room temperature $T$ (in ºC) according to the expression $C(T) = |T^2 - 40T + 375|$. For the product to be competitive in the market, the unit cost must not exceed 25 euros.
The figure shows the graph of $C(T)$ and the line $C = 25$.
Solve the inequality $|T^2 - 40T + 375| \leq 25$ and determine the temperature ranges allowed for manufacturing.