Mathematics · Expressions, equations and inequalities · Grade 8

Inequalities: ray and interval

A lift carries 630 kg. How many 75 kg people can get in?

The situation

The goods lift of a multi-storey building carries 630 kg. How many people of 75 kg on average can get in at once? The answer is not a single number: any number from 0 to 8 will do — this is an inequality, not an equation. “No more than 20 GB a month” or “a height that differs from 170 cm by 5 cm” are inequalities too. We will learn to solve them with a ray, an interval and a distance on the number line.

Topic

Numerical inequalities and their properties, inequalities in one unknown and systems of them, intervals, the modulus. The limit of a goods lift and a mobile tariff: “not more than” becomes a ray and an interval on the number line; |x − a| is a distance

Key ideas

Numerical inequality
Number interval
Modulus — distance

Formulas

a > b \Leftrightarrow a - b > 0

The definition of a numerical inequality: the number $a$ is greater than $b$ when the difference is positive, and on the line $a$ is further to the right.

a > b \Rightarrow a + c > b + c; a c > b c (c > 0), a c < b c (c < 0)

The same number can be added to both sides; when we multiply by a negative number, the inequality sign reverses.

a > b, c > d \Rightarrow a + c > b + d; a > b > 0, c > d > 0 \Rightarrow a c > b d

Inequalities of the same sense can be added term by term; if the numbers are positive, they can be multiplied as well.

| x - a | \leq h \Leftrightarrow a - h \leq x \leq a + h

$| x - a |$ is the distance from $x$ to $a$ ; the points whose distance does not exceed $h$ form the closed interval $[a - h; a + h]$ .

Theory

Numerical inequality

To compare two numbers we take their difference: if $a - b > 0$ , then $a > b$ ; if $a - b < 0$ , then $a < b$ ; if the difference is zero, the numbers are equal. On the number line the larger number is further to the right: $5 > 2$ , $- 2 > - 5$ . In the first step, move the two points and look at the arrow showing the difference — if it points to the right, the difference is positive.

Basic properties

If we add the same number to both sides (or subtract it), the points shift together along the line and the order does not change: $a > b \Rightarrow a + c > b + c$ . If we multiply by a positive number, the distances stretch and the order is kept. If we multiply by a negative number, the points are flipped about 0 and the order is reversed: $5 > 2$ , but $- 10 < - 4$ . That is why we reverse the sign when we divide an inequality by a negative number. Also: if $a > b$ and $b > c$ , then $a > c$ (transitivity).

Addition, multiplication, powers

Inequalities of the same sense can be added term by term: if $a > b$ and $c > d$ , then $a + c > b + d$ — put two longer segments end to end and the total is longer too. For positive numbers we can multiply as well: $a c > b d$ — a rectangle with both the greater width and the greater height has the greater area. In particular, if $a > b > 0$ , then $a^{n} > b^{n}$ . For negative numbers this fails: $- 3 < - 2$ , but ${(- 3)}^{2} = 9 > 4$ .

An inequality in one unknown and a system

We solve an inequality like an equation: we move terms across and divide by the coefficient — only when dividing by a negative number the sign reverses. The solution is a ray on the number line: for $x \geq 2$ it is $[2; + \infty)$ with a closed (black) endpoint (included); for $x < 5$ it is $(- \infty; 5)$ with an open (white) endpoint (not included). A system means both inequalities at once: the common part of the rays, for example $[2; 5)$ . If there is no common part, there is no solution ( $⌀$ ). In the lift, $75 n \leq 630$ gives $n \leq 8.4$ , and since $n$ is a natural number, at most 8 people.

Modulus — distance

$| x - a |$ is the distance from $x$ to $a$ on the number line. So the solution of $| x - a | = h$ is the two points $a \pm h$ at distance $h$ from $a$ ; $| x - a | \leq h$ is the closed interval $[a - h; a + h]$ between them; $| x - a | \geq h$ is two rays. On the graph, $y = | x - a |$ is a “V” with its vertex at $a$ ; the line $y = h$ cuts it at two points, and the closed interval is the part of the V below $h$ . In a double inequality we do the same operation on all three parts; with two moduli, the line is divided by their zeros into zones.

About the model

The coefficients in the steps are whole numbers, but the answers can be fractions — on the line they are placed exactly, and in writing they appear as a fraction in lowest terms or as a decimal approximation with “≈”. In the lift problem the “average of 75 kg” is assumed (lift standards calculate it the same way); real passengers differ. The number line is shown from $- 12$ to $12$ — solutions outside this range do not fit on the line, but they do appear in the formula.

Did you know?

The signs “<” and “>” were introduced by the English mathematician Thomas Harriot; they appeared in his “Artis analyticae praxis”, published in 1631 after his death. The signs “≤” and “≥” were used by the French scientist Pierre Bouguer in 1734.
In the European lift standard (EN 81), the load capacity of the car is tied to the number of passengers at 75 kg each: a 630 kg lift is designed for 8 passengers and a 1 000 kg lift for 13.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Monomials and polynomials — Grade 7
Special products (algebraic identities) — Grade 7
Algebraic fractions — Grade 7
The linear equation: the balance — Grade 7
Systems of equations — Grade 7
The quadratic equation: the parabola — Grade 8