Nota Mathematics Tingkatan 4 Bab 6

Inequalities describe the relationship between two quantities that are not equal. (p.156)
Symbols: > greater than; < less than; ≥ greater than or equal to; ≤ less than or equal to. (p.156)
Linear equation in two variables: ax + by = c (a, b, c constants); the highest power of a variable is 1. (p.156)
Representing a situation: let x and y be the variables, e.g. books at RM14 and RM9 with at most RM100 → 14x + 9y ≤ 100. (p.157)
Multiply both sides by the same number to clear fractions or decimals, e.g. 12.50k + 18.50b ≤ 150 → 25k + 37b ≤ 300. (p.157)
General form of a straight line: y = mx + c (m = gradient, c = y-intercept). Gradient of y = h is zero; gradient of x = k is undefined. (p.158)
A straight line divides the Cartesian plane into region above/below (for y = mx + c) or region left/right (for x = k). (p.158)

Points on the line satisfy y = mx + c; points in the region above satisfy y > mx + c; points in the region below satisfy y < mx + c. (p.159)
To test a point without drawing: substitute x into mx + c and compare y with the result (=, > or <). (p.160)
For y = h: region above is y > h; region below is y < h. (p.161)
For x = k: region to the right is x > k; region to the left is x < k. (p.162)
Use a dashed line for > and < (points on the line not included); use a solid line for ≥ and ≤ (points on the line included). (p.158)
To draw a region: convert the inequality to an equation, draw the line (two points are enough), then shade the correct region. (p.163)
Make y the subject first so that the region (above/below) is marked correctly; e.g. x − y > 4 → y < x − 4. (p.164)

A system of linear inequalities is a combination of two or more linear inequalities. (p.165)

Words to inequality (p.165)

Situation	Linear inequality
y is greater than x	y > x
y is less than x	y < x
y is not less than x	y ≥ x
y is not more than x	y ≤ x
y is at least k times x	y ≥ kx
y is at most k times x	y ≤ kx
Maximum of y is k	y ≤ k
Minimum of y is k	y ≥ k
Sum of x and y is greater than k	x + y > k
Difference between y and x is less than k	y − x < k
y is more than x by at least k	y − x ≥ k

x ≥ 0 and y ≥ 0 are included when x and y represent quantities. (p.166)
To find which region satisfies a system, substitute at least one point from each region into every inequality. (p.167)

Steps: (a) mark the region of each inequality with different markings; (b) identify the common region; (c) shade it completely and only within the common region. (p.169)
y ≥ 0 is the region above the x-axis (y = 0); x ≥ 0 is the region to the right of the y-axis (x = 0). (p.169)
Convert each inequality to an equation, draw all the lines, identify the common region, then shade it. (p.170)
To write inequalities for a shaded region: find each boundary line’s equation, use dashed/solid line to choose the sign, and shade above/below or left/right to choose the direction. (p.171)
Linear equations: x-axis is y = 0; y-axis is x = 0. (p.171)
Gradient m = (y₂ − y₁) / (x₂ − x₁). (p.172)
Graphs of inequalities for quantities are drawn in the first quadrant only. (p.173)
Systems of linear inequalities are the basis in business for obtaining maximum profit with minimum cost using limited manpower, raw materials and finance. (p.173)
From the graph, read the minimum/maximum value of one variable for a given value of the other; a point outside the shaded region does not meet the conditions. (p.173)
Problem solving steps: define variables, write the inequalities, draw and shade the common region, then read values from the graph. (p.174)

Chapter 6: Linear Inequalities in Two Variables