First-degree inequalities
So far, each equation has had a specific number of solutions. But many real-life questions are not "exactly how much?" but "from what point onwards?" or "up to how much at most?". That is what inequalities are for: their solution is not a single number, but a whole interval of numbers.
What an inequality is
The inequality $2x-1>5$ has as its solution all the numbers greater than $3$: $4$, $3{,}1$, $100$... infinitely many values. This solution is written as an interval: $(3, +\infty)$.
The figure shows the solution set of an inequality on the number line. a) Express it as an interval. b) Express it using an inequality. c) Write a linear inequality…
The figure shows the solution set of an inequality on the number line.
- a) Express it as an interval. b) Express it using an inequality. c) Write a linear inequality whose solution is exactly that set.
Check whether the values $x = 2$ and $x = 8$ are solutions to the inequality $5(x - 1) - 2x < 10$, without solving it beforehand. Then, solve it and confirm your answer.
Check whether the values $x = 2$ and $x = 8$ are solutions to the inequality $5(x - 1) - 2x < 10$, without solving it beforehand. Then, solve it and confirm your answer.
Solving a first-degree inequality
It is solved in the same way as an equation: by moving terms from one side to the other. But there is a new rule, and it is very important:
Solve the inequality $3x - 5 < 7$ and express its solution set in interval notation.
Solve the inequality $3x - 5 < 7$ and express its solution set in interval notation.
Solve the inequality $5(x - 2) \leq 3x + 4$.
Solve the inequality $5(x - 2) \leq 3x + 4$.
A classmate claims: "to solve $-3x > 6$, I divide both sides by $-3$ and get $x > -2$". Decide whether their statement is true or false, explain the error if there is…
A classmate claims: "to solve $-3x > 6$, I divide both sides by $-3$ and get $x > -2$". Decide whether their statement is true or false, explain the error if there is one, and give the correct solution.
Double inequalities and systems of inequalities
Sometimes a quantity has to lie between two values at the same time, or satisfy two conditions.
Solve the double inequality $-3 \leq 2x + 1 < 7$ and express the solution as an interval.
Solve the double inequality $-3 \leq 2x + 1 < 7$ and express the solution as an interval.
Solve the system of inequalities $\begin{cases} 2x - 3 < 5 \\ x + 4 \geq 1 \end{cases}$ and represent the solution on the number line.
Solve the system of inequalities $\begin{cases} 2x - 3 < 5 \\ x + 4 \geq 1 \end{cases}$ and represent the solution on the number line.
The figure shows two intervals $A$ and $B$ represented on the number line. Write, as an interval, the set of values of $x$ that belong to both $A$ and $B$ at the same…
The figure shows two intervals $A$ and $B$ represented on the number line.
Write, as an interval, the set of values of $x$ that belong to both $A$ and $B$ at the same time, that is, the solution to the system formed by the two corresponding inequalities.
Problems with inequalities
Ainhoa wants to get a mean score of at least $6{,}5$ across the three mathematics exams this term. In the first two, she scored $5{,}6$ and $7{,}2$. What minimum mark…
Ainhoa wants to get a mean score of at least $6{,}5$ across the three mathematics exams this term. In the first two, she scored $5{,}6$ and $7{,}2$. What minimum mark does she need in the third exam?
Two phone companies offer the following tariffs: Company A charges a monthly fee of $15\ \text{€}$ plus $0{,}03\ \text{€}$ per minute, and Company B charges no fee but…
Two phone companies offer the following tariffs: Company A charges a monthly fee of $15\ \text{€}$ plus $0{,}03\ \text{€}$ per minute, and Company B charges no fee but $0{,}08\ \text{€}$ per minute. a) Write the monthly cost for each company as a function of the minutes $m$ used. b) For what monthly usage do both cost the same? c) For what values of $m$ is Company A cheaper?
To move the school's equipment, two van hire companies are compared: KmCero charges $45\ \text{€}$ per day plus $0{,}25\ \text{€}$ per kilometre, and RutaPlana charges…
To move the school's equipment, two van hire companies are compared: KmCero charges $45\ \text{€}$ per day plus $0{,}25\ \text{€}$ per kilometre, and RutaPlana charges $70\ \text{€}$ per day plus $0{,}10\ \text{€}$ per kilometre. a) Write the cost for one day's hire for each company as a function of the distance in kilometres $x$ driven. b) If it is expected to drive $120$ km in one day, which company is better value? c) Above how many kilometres per day is RutaPlana cheaper? d) A teacher claims: "we always win with KmCero because its fixed price is lower". Explain, using your results, why they are wrong.