# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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(b) x < 5.

(c) x + y > 5.

(d) x + y < 8.

Solution. To graph an inequation, we first draw the line graph and then shade the region that represents the inequality.

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250 INEQUALITY

y x> 3 y x < 5

0 3 X 0 5 X

(a) (b)

(a) Draw the graph of the straight line x = 3 and shade all the region to the right of the line, and include the line itself (see figure a). This shaded region represents the inequality x greater than or equal to 3. The line itself is included in the shading by drawing the line in full and not dashed.

(b) Draw the graph of the straight line x = 5 and shade all the region which is to the left of the line, including the line itself (see figure b). This shaded region represents the inequality x less than or equal to 5.

(c) Draw the graph of the straight line x + y = 5, but draw it dashed, because the line itself is not included in the region, since the inequality is > and not >. Shade the “greater than” region, which is the entire region above the line, not including the line (see figure c).

(d) Draw the graph of the straight line x + y = 8, and draw it in full. Shade the “less than or equal to” region, which is the entire region below the line, including the line itself (see figure d).

(d)

These four graphs of inequations will be combined onto a single set of axes to solve the next problem.

Example 2. The Crabtrees family is planning a holiday at Hotel Cheapo, which offers discounts for groups of people. The rales laid down by the hotel, in order to qualify for a discount, are these:

1. There must be more than five people in the group, but no more than eight.

2. There must be at least three adults, but no more than five.

What size groups can go to Hotel Cheapo for a holiday at a discount price?

INEQUALITY 251

Solution. Let the number of adults in the group be x and the number of children be y. The inequalities that fit the information in this problem are the same as the inequalities that have just been drawn:

x > 3 There must be at least three adults

x < 5 There must be no more than five adults

x + y > 5 There must be more than five people

x + y < 8 There must be no more than eight people

These four graphs will all be drawn on one set of axes. To enable us to concentrate on the intersections of the four inequalities, we will shade the “non” region for each inequality, which leaves the region we want unshaded (see figure e).

(e)

There are nine points in the unshaded region, and any of those points represents a group of people that qualify for the discount. The coordinates of the nine points are

(3, 5), (3, 4), (3, 3), (4, 4), (4, 3), (4, 2), (5, 3), (5, 2), (5, 1)

The first coordinate represents the number of adults in the group and the second coordinate represents the number of children in the group. For example, the point (3, 5) is a group of three adults and five children that qualifies for the discount. Note that points on the dashed line are not in the region. This process of solving a problem by graphs of inequalities is called linear programming.

References: Gradient-Intercept Form, Graphs, Greater Than, Inequations, Parabola, Quadratic Graphs.

252 INEQUATIONS

INEQUATIONS

An inequation is similar to an equation, except that instead of an equal sign joining two expressions, an inequality sign joins them. The equation 2x + 5 = 3 can be written as an inequation by replacing = by < to give 2x + 5 < 3. Inequations can be solved using skills similar to those for solving equations (refer to the entry Balancing an Equation). There is one important difference between balancing equations and balancing inequations. When multiplying (or dividing) both sides of an inequation by a negative number it is necessary to turn around the inequality sign before continuing with the solution. This process is demonstrated in the second example. You may also be interested in graphing the solution to an inequation on a number line, and this is demonstrated in the first example.

Example 1. Solve yx 2 > 1.

Solution. Write

\x -2 > -1

|x > 1 Adding 2 to both sides of the inequation

x > 1.5 Dividing both sides by |

o—►

* i i_______i____i___i____>

-2-1 0 1 2

The solution is graphed on the number line as in the accompanying figure. Note the open circle above 1.5, which indicates that 1.5 is not included in the graph of the answer. Refer to the entry Greater Than.

Example 2. Solve 1 - 2x > 5.

Solution. Write

1 — 2x > 5

—2x>4 Subtracting 1 from both sides of inequation

The next step is to divide both sides of the inequation by —2, and this process includes turning around the inequality. Write

x < —2 The sign > is turned around and becomes <

References: Balancing an Equation, Greater Than, Inequality, Number Line.

INSCRIBE 253

INFINITE

If something is infinite, then it is not finite. Infinity is a word used to describe the size of a quantity (which may be a set, sequence, or group) which has no limit. Its size has no restriction, it is boundless. For example, the set of positive even numbers {2,4,6,8,...} is infinite, because there is no last number in the sequence. In geometry we say that a line has infinite length, whereas a line segment has a finite length.

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