Inequalities and intervals – Equations and Inequalities (Senior 4 Basic Mathematics) | Ganzaa

Equations and Inequalities

Lessons

Senior 4 · Basic Mathematics

Equations and Inequalities

5 lessons

Lesson 2 of 5

Inequalities and intervals

25 min

"At most", "at least", "under"

A lift notice says "maximum 8 persons or 630 kg". A bus company sells half-price tickets to children under 12. A scholarship needs an average mark of at least 70%. None of these is an equation: each describes a whole range of allowed values. The total mass in the lift can be 400 kg or 629.5 kg or exactly 630 kg, but not 631 kg. Statements like these are inequalities, and their solutions are written as intervals of the number line.

Words	Symbol	Example
is less than, is under, is below	$< \lt$	age $a \lt 12$
is at most, does not exceed, maximum	$\leq \le$	mass $m \le 630$
is greater than, is more than, is above	$> \gt$	profit $p \gt 0$
is at least, is no less than, minimum	$\geq \ge$	mark $x \ge 70$

Definition

Inequality: a statement that one expression is less than ( $< \lt$ ), less than or equal to ( $\leq \le$ ), greater than ( $> \gt$ ) or greater than or equal to ( $\geq \ge$ ) another. Its solution set is the set of all values of the unknown that make it true.

Example: the solution set of $x + 3 \gt 5$ is every real number greater than 2: $x \gt 2$ .

Important

Rules for solving inequalities

Adding or subtracting the same number on both sides keeps the inequality sign.
Multiplying or dividing both sides by a positive number keeps the sign.
Multiplying or dividing both sides by a negative number reverses the sign.

Why the last rule: $-2 \lt 3$ is true, but after multiplying by $- 1$ we get $2$ and $- 3$ , and $2 \gt -3$ . Multiplying by a negative number reflects the number line, so the order flips.

Intervals

An interval is the set of all real numbers between two end points. A square bracket facing the number means the end point is included; a bracket facing away (or a round bracket) means it is not. On a number line a filled dot means "included", a hollow dot "not included". The symbol $\infty$ is not a number, so its end is always open.

Inequality	Interval	Also written
$a \le x \le b$	$[a, b]$ (closed)
$a \lt x \lt b$	$] a, b [$ (open)	$(a, b)$
$a \le x \lt b$	$[a, b [$	$[a, b)$
$x \ge a$	$[a, +\infty[$	$[a, \infty)$
$x \lt b$	$]-\infty, b[$	$(-\infty, b)$

Intervals are sets, so they can be combined. The intersection $A \cap B$ contains the numbers in both; the union $A \cup B$ contains the numbers in at least one:

Worked example 1: solving linear inequalities

Solve and write each solution set as an interval: (a) $3 - 2x \le 11$ (b) $\frac{x}{2} - 1 \gt \frac{x + 1}{3}$ (c) $-3 \lt 2x + 1 \le 7$ .

Solution

$-2x \le 8$ . Divide by $- 2$ and reverse the sign: $x \ge -4$ . $S = [-4, +\infty[$ .
Multiply by 6 (positive): $3x - 6 \gt 2x + 2$ , so $x \gt 8$ . $S = ]8, +\infty[$ .
Do the same to all three parts: subtract 1, $-4 \lt 2x \le 6$ ; divide by 2, $-2 \lt x \le 3$ . $S =] - 2, 3]$ .

Worked example 2: two conditions at once

Find the values of $x$ that satisfy both $2x - 1 \ge -5$ and $x + 4 \lt 7$ .

Solution

The first gives $x \ge -2$ , that is $[-2, +\infty[$ ; the second gives $x \lt 3$ , that is $]-\infty, 3[$ . "Both" means the intersection:

S = [-2, +\infty[ \,\cap\, ]-\infty, 3[ \, = [-2, 3[ .

Worked example 3: inequalities in everyday decisions

(a) Keza has 12 000 Frw. She must buy a book for 3 000 Frw, and wants to spend the rest on pens at 450 Frw each. How many pens can she buy? (b) Mobile data plan A costs 2 000 Frw a month plus 20 Frw per MB; plan B costs 50 Frw per MB with no monthly fee (illustrative prices). For which monthly use is plan A cheaper?

Solution

Let $p$ be the number of pens: $3000 + 450p \le 12\,000$ , so $450p \le 9000$ and $p \le 20$ . She can buy at most 20 pens ( $p$ is a whole number from 0 to 20).
Let $m$ be the number of MB. A is cheaper when $2000 + 20m \lt 50m$ , that is $2000 \lt 30m$ , so $m \gt 66.\overline{6}$ . Plan A is cheaper for 67 MB or more a month; plan B for 66 MB or less.

Warning

The most common error is forgetting to reverse the sign when dividing by a negative number: from $-3x \lt 12$ the answer is $x \gt -4$ , not $x \lt -4$ . Check with one value: $x = 0$ is in $x \gt -4$ , and $-3(0) = 0 \lt 12$ is true. ✓ Also take care at the ends: "at most 20" includes 20, so the bracket is closed.

Tip

After solving, test one number inside your interval and one outside it in the original inequality. The inside number must make it true and the outside one false. It takes ten seconds and catches a reversed sign every time.

Mathispower4u: Number Line Intervals Using Inequalities and Interval Notation. Watch how open and closed dots match round and square brackets.

✦

Practice

Try each one before opening the answers. They get harder as you go.

Write as an interval and draw on a number line: (a) $-1 \lt x \le 5$ (b) $x \ge 2.5$ (c) $x \lt 0$ .
Write as an inequality: (a) $[- 3, 7 [$ (b) $]-\infty, -1]$ .
Solve $5x - 3 \lt 2x + 9$ .
Solve $4 - 3x \ge 10$ .
Solve $1 \le \frac{3 - x}{2} \lt 4$ and give the solution as an interval.
Find $[-2, 5] \cap ]1, 8]$ and $[-2, 5] \cup ]1, 8]$ .
A hall costs 50 000 Frw to hire and 2 500 Frw per guest for food. A family has a budget of 300 000 Frw. How many guests can they invite at most?

Show answersHide answers

(a) $] - 1, 5]$ : hollow dot at $- 1$ , filled at 5 (b) $[2.5, +\infty[$ (c) $]-\infty, 0[$ .
(a) $-3 \le x \lt 7$ (b) $x \le -1$ .
$3x \lt 12$ , $x \lt 4$ : $]-\infty, 4[$ .
$-3x \ge 6$ , $x \le -2$ : $]-\infty, -2]$ .
$2 \le 3 - x \lt 8$ , $-1 \le -x \lt 5$ , so $-5 \lt x \le 1$ : $] - 5, 1]$ .
$] 1, 5]$ and $[- 2, 8]$ .
$50\,000 + 2500g \le 300\,000$ , $g \le 100$ : at most 100 guests.

Quick check

Which interval is the solution set of 7 − 2x > 1?

Key points

Solve an inequality like an equation, but reverse the sign when you multiply or divide by a negative number.
$[a, b]$ includes its ends, $] a, b [$ does not; $\pm\infty$ is always open.
"Both conditions" is an intersection; "either condition" is a union.
In word problems, check whether the unknown must be a whole number.

Sources and credits

Written by Ganzaa: the explanations, the diagrams, the worked examples, the practice items and the questions. All prices are illustrative.
Video: Number Line Intervals Using Inequalities and Interval Notation, Mathispower4u, embedded from YouTube.

1"At most", "at least", "under"

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"At most", "at least", "under"

Intervals