Simultaneous Linear Inequalities

Lessons

Senior 3 · Mathematics

Simultaneous Linear Inequalities

4 lessons

Lesson 1 of 4

Compound statements and simultaneous inequalities in one unknown

25 min

Between 2 °C and 8 °C

A health centre in Nyagatare stores its vaccines in a refrigerator. Most vaccines must be kept between $2\,^\circ\mathrm{C}$ and $8\,^\circ\mathrm{C}$ : colder and some are damaged by freezing, warmer and they slowly lose their power. If $t$ is the temperature in degrees Celsius, the nurse needs two statements to be true at the same time:

t \ge 2 \quad\text{and}\quad t \le 8, \qquad\text{written together as}\qquad 2 \le t \le 8.

The safe range for the vaccine fridge. A filled dot means the end value is allowed.

In S1 you solved one linear inequality at a time. In this lesson you join inequalities with the words and and or, and find the numbers that satisfy both conditions at once, exactly as the nurse does.

Definition

Solution set of an inequality: the set of all values of the unknown that make the inequality true.

Notation: in set-builder form, $\{x : x > 2\}$ , read "the set of all $x$ such that $x$ is greater than 2". In interval form, $]2, +\infty[$ . Some books write the same interval as $(2, \infty)$ .

Example: the solution set of $2 \le t \le 8$ is $\{t : 2 \le t \le 8\} = [2, 8]$ . A square bracket facing the number, $[2$ , means 2 is included; a bracket facing away, $] 2$ , means 2 is left out.

Showing an inequality on a number line

Inequality	Interval	End point on the number line
$x > a$	$]a, +\infty[$	open dot at $a$ , arrow to the right
$x \ge a$	$[a, +\infty[$	filled dot at $a$ , arrow to the right
$x < a$	$]{-}\infty, a[$	open dot at $a$ , arrow to the left
$x \le a$	$]{-}\infty, a]$	filled dot at $a$ , arrow to the left
$a < x \le b$	$] a, b]$	open dot at $a$ , filled dot at $b$ , segment between

Infinity is not a number, so the bracket at $\pm\infty$ always faces away.

Important

Solving a linear inequality. You may add or subtract the same number on both sides, and multiply or divide both sides by the same positive number. If you multiply or divide by a negative number, you must reverse the inequality sign.

Why: $- 2 < 3$ is true. Multiply both sides by $- 1$ : the numbers become $2$ and $- 3$ , and $2 > - 3$ . Multiplying by a negative number reverses the order of numbers on the number line, so the sign must turn round too.

Definition

Compound statement: two statements joined by and or by or.

" $P$ and $Q$ " is true only when both $P$ and $Q$ are true. Its solution set is the intersection $A \cap B$ of the two solution sets.
" $P$ or $Q$ " is true when at least one of them is true. Its solution set is the union $A \cup B$ .

Example: " $x > 1$ and $x < 4$ " has solution set $] 1, 4 [$ ; " $x < 1$ or $x > 4$ " has solution set $]{-}\infty, 1[\; \cup\; ]4, +\infty[$ . You met $\cap$ and $\cup$ in Unit 1 on sets.

Definition

Simultaneous inequalities (a system of inequalities): two or more inequalities in the same unknown that must all be true at the same time. They are joined by "and", so the solution set is the intersection of the separate solution sets.

Notation: a system is often written with a brace, $\begin{cases} 3x - 1 > 5 \\ 2x + 3 \le 13 \end{cases}$

Example: $2 \le t \le 8$ is the short form of the system $t \ge 2$ and $t \le 8$ .

Worked example 1: a system of two inequalities

Solve the system $\begin{cases} 3x - 1 > 5 \\ 2x + 3 \le 13 \end{cases}$ and list the whole numbers that satisfy it.

Solution Solve each inequality on its own.

3x - 1 > 5 \;\Rightarrow\; 3x > 6 \;\Rightarrow\; x > 2 \qquad\qquad 2x + 3 \le 13 \;\Rightarrow\; 2x \le 10 \;\Rightarrow\; x \le 5

Draw both on one number line and look for the overlap.

Both conditions hold only in the green overlap: 2 is left out (open dot), 5 is included (filled dot).

So $2 < x \le 5$ , that is, $S =] 2, 5]$ . The whole numbers in it are $3, 4, 5$ .

Check $x = 5$ : $3 (5) - 1 = 14 > 5$ ✓ and $2(5) + 3 = 13 \le 13$ ✓.

Worked example 2: a double inequality

Solve $-5 \le 3 - 2x < 7$ .

Solution A double inequality is two inequalities at once, so do the same operation to all three parts.

-5 \le 3 - 2x < 7 \;\xrightarrow{\;-3\;}\; -8 \le -2x < 4 \;\xrightarrow{\;\div(-2)\text{, reverse}\;}\; 4 \ge x > -2.

Written with the smaller number first: $-2 < x \le 4$ , so $S =] - 2, 4]$ .

Check the ends. $x = 4$ : $3 - 8 = - 5$ , and $-5 \le -5$ ✓, so 4 is included. $x = - 2$ : $3 + 4 = 7$ , but $7 < 7$ is false, so $- 2$ is left out ✓.

Worked example 3: "or", and a system with no solution

Solve (a) $x + 4 < 2$ or $3x \ge 9$ (b) $2 x > 8$ and $x - 1 < 2$ .

Solution (a) $x + 4 < 2 \Rightarrow x < -2$ , and $3x \ge 9 \Rightarrow x \ge 3$ . With "or", a number is a solution if it satisfies either one:

With “or”, every marked number is a solution: the two rays together.

$S = {]}{-}\infty, -2[\; \cup\; [3, +\infty[$ . Numbers such as 0 or 1 satisfy neither, so they are not solutions.

(b) $2x > 8 \Rightarrow x > 4$ , and $x - 1 < 2 \Rightarrow x < 3$ . No number is both greater than 4 and less than 3:

With “and”, a solution must lie on both rays. These rays never overlap.

The system has no solution: $S = \varnothing$ .

Warning

Three common mistakes

Forgetting to reverse. From $- 2 x < 4$ the answer is $x > - 2$ , not $x < - 2$ . Check with $x = 0$ : $0 < 4$ is true, and 0 is greater than $- 2$ .
Squeezing an "or" into one double inequality. " $x < - 2$ or $x \ge 3$ " must never be written $3 \le x < -2$ : that says $3 < - 2$ , which is false, so it describes no numbers at all.
Losing the end points. Decide for each end separately whether it is included: $\leq \le$ and $\geq \ge$ give a filled dot and a square bracket facing the number.

Worked example 4: exercise books for the term

Keza needs at least 8 exercise books for the term. Each costs 400 Frw, and she may spend at most 5,000 Frw (illustrative prices). How many books can she buy?

Solution Let $n$ be the number of books. The two conditions are

n \ge 8 \qquad\text{and}\qquad 400n \le 5000 \;\Rightarrow\; n \le \frac{5000}{400} = 12.5.

So $8 \le n \le 12.5$ . But $n$ counts books, so it must be a whole number: $n \in \{8, 9, 10, 11, 12\}$ . She can buy from 8 to 12 books.

A vaccine refrigerator fitted with numbered temperature probes (the thin wires) during a study by the US National Institute of Standards and Technology of how well vaccines keep their temperature. The fridge is doing its job only while every probe reads between 2 °C and 8 °C. Photo: NIST, public domain, via Wikimedia Commons

Khan Academy: Compound inequalities. Watch how each inequality is solved on its own, and how the word “and” or “or” decides which part of the number line is kept.

Tip

"And" means overlap; "or" means everything marked. Draw both solution sets on one number line, one above the other. For "and", keep only the part where they overlap. For "or", keep every part that is marked.

✦

Practice

Solve each one, show it on a number line and give the solution set. The last three are harder.

Write in set-builder and interval notation: (a) $x \ge -1$ (b) $- 3 < x < 2$ .
$5 - 2 x > 11$
$\begin{cases} 2x + 1 \ge -3 \\ x - 4 < 0 \end{cases}$
$-1 < 2x + 3 \le 9$ . List the integer solutions.
$4 \le 10 - 3x < 16$
$x - 3 > 1$ or $2x + 5 \le 1$
$3 x - 2 > 7$ and $5 - x \ge 4$
A rectangle is 5 cm wide and $x$ cm long. Its perimeter must be at least 24 cm and its area less than $60\ \mathrm{cm^2}$ . Find the possible values of $x$ .

Show answersHide answers

(a) $\{x : x \ge -1\} = [-1, +\infty[$ (b) $\{x : -3 < x < 2\} = {]}{-}3, 2[$ .
$- 2 x > 6$ , so $x < - 3$ : $]{-}\infty, -3[$ .
$x \ge -2$ and $x < 4$ : $-2 \le x < 4$ , that is, $[- 2, 4 [$ .
$-4 < 2x \le 6$ , so $-2 < x \le 3$ ; integers $- 1, 0, 1, 2, 3$ .
$-6 \le -3x < 6$ ; divide by $- 3$ and reverse: $2 \ge x > -2$ , so $] - 2, 2]$ .
$x > 4$ or $x \le -2$ : $]{-}\infty, -2]\; \cup\; ]4, +\infty[$ .
$x > 3$ and $x \le 1$ : no number does both, so $S = \varnothing$ .
$2(x + 5) \ge 24 \Rightarrow x \ge 7$ ; $5x < 60 \Rightarrow x < 12$ . So $7 \le x < 12$ .

Sources and credits

Written by Ganzaa.
Vaccine storage range: Controlled temperature chain (CTC), World Health Organization, which notes that almost all vaccines are licensed for storage in the traditional cold chain of $2\,^\circ\mathrm{C}$ to $8\,^\circ\mathrm{C}$ .
Photo: Vaccine; Temperature; Refrigerator (5941076542).jpg, National Institute of Standards and Technology, Wikimedia Commons, public domain. Used unchanged.
Video: Compound inequalities, Khan Academy, embedded from YouTube.

The worked examples, number lines, practice items and quiz questions were written by Ganzaa. Prices are illustrative.

1Between 2 °C and 8 °C

2Showing an inequality on a number line

Between 2 °C and 8 °C

Showing an inequality on a number line