Some inequalities are awkward to untangle with algebra but obvious in a picture. Draw the two sides as two graphs on one grid, then just look at which one is on top. The answer is read off the x-axis.
1. Above, below, and where they meet
When the algebra is hard, both sides of the inequality are graphed on the same grid and the sign of the inequality decides what to look for. The solution is then marked on the x-axis from the shape of the graphs.
Three rules. The solutions of f(x) = g(x) are the x-coordinates of the points where y = f(x) and y = g(x) cross. The solution set of f(x) < g(x) is the x-values where the graph of f lies below the graph of g. And f(x) > g(x) holds where f lies above. Slide x and change the sign below.
two graphsslide x, change the sign
Which graph is on top?
Figure 1.8.1 in the book shows exactly these two graphs: f(x) = 4 − 2|x| is an upside-down V with its apex at (0, 4), and g(x) = 2x² is the parabola through the origin. They cross at (−1, 2) and (1, 2). So f = g at x = −1 and x = 1; f < g on (−∞, −1) ∪ (1, ∞); f > g on (−1, 1); and f ≥ g on [−1, 1], because the two meeting points count too. Now find them in the picture yourself.
read the graphtap a dot
Where is it below?
The two outer stretches. Here the blue graph f is below and g is above. So f < g on (−∞, −1) ∪ (1, ∞).
The middle. From −1 to 1, f is on top. So f > g on (−1, 1).
The meeting points. (−1, 2) and (1, 2). f = g at x = −1 and 1; add the stretch between them and f ≥ g is [−1, 1].
All three seen. Below means less, above means more.
2. The book's three examples
Example 15: the graph of f(x) = x/2 + 1 is a straight line through (−2, 0) and (0, 1). f(x) ≤ 0 means the line is on or below the x-axis, which happens for x ∈ (−∞, −2]. The solution is x ≤ −2.
Example 16: for −3 ≤ 4 − 7x < 18 draw three graphs, y₁ = −3, y₂ = 4 − 7x and y₃ = 18. For x ∈ (−2, 1], y₂ stays between y₁ and y₃. The solution is −2 < x ≤ 1.
Example 17: x² − 4x + 3 < 0, that is (x − 3)(x − 1) < 0. The graph shows the expression is positive for x < 1 or x > 3 and negative for x ∈ (1, 3). The solution is 1 < x < 3.
The book's graph exercises come out by the same rule: −1 < 5 − 2x ≤ 9 has solution −2 ≤ x < 3; x² − 5x > 6 has solution x < −1 or x > 6; and x/(x − 2) ≤ 2 has solution x < 2 or x ≥ 4. Five of these are graphed below; open them one at a time.
solve by graphpick an inequality
Read the solution
The book draws its graphs on graph paper so the crossing points land exactly. Without graph paper you can still do it: draw a number line freehand and mark the solution on it.
3. Check yourself
Three questions in the exam's style. Ask first which graph is on top.
check yourselfpick one answer
Question 1
Which part of the graphs gives the solution set of f(x) < g(x)?
Right. f(x) < g(x) means f is smaller, i.e. its graph is lower.
where the graph of f is above that of g does not fit here; the right answer is where the graph of f is below that of g. f(x) < g(x) means f is smaller, i.e. its graph is lower.
where the graphs intersect does not fit here; the right answer is where the graph of f is below that of g. f(x) < g(x) means f is smaller, i.e. its graph is lower.
where the graph of f is above the x-axis does not fit here; the right answer is where the graph of f is below that of g. f(x) < g(x) means f is smaller, i.e. its graph is lower.
check yourselfpick one answer
Question 2
The graphs meet at (−1, 2) and (1, 2) and for −1 < x < 1 the graph of f lies above that of g. What is the solution of f(x) ≥ g(x)?
(−1, 1) does not fit here; the right answer is [−1, 1]. f > g has solution (−1, 1) and f = g has x = ±1; so f ≥ g has solution [−1, 1].
(−∞, −1) ∪ (1, ∞) does not fit here; the right answer is [−1, 1]. f > g has solution (−1, 1) and f = g has x = ±1; so f ≥ g has solution [−1, 1].
Right. f > g has solution (−1, 1) and f = g has x = ±1; so f ≥ g has solution [−1, 1].
{−1, 1} does not fit here; the right answer is [−1, 1]. f > g has solution (−1, 1) and f = g has x = ±1; so f ≥ g has solution [−1, 1].
check yourselfpick one answer
Question 3
How is the solution of f(x) = g(x) obtained from the graphs?
from the y-intercepts does not fit here; the right answer is from the x-coordinates of the intersections of y = f(x) and y = g(x). At an intersection f(x) = g(x), so the x-coordinate of that point is the solution.
Right. At an intersection f(x) = g(x), so the x-coordinate of that point is the solution.
from the slopes does not fit here; the right answer is from the x-coordinates of the intersections of y = f(x) and y = g(x). At an intersection f(x) = g(x), so the x-coordinate of that point is the solution.
from the origin does not fit here; the right answer is from the x-coordinates of the intersections of y = f(x) and y = g(x). At an intersection f(x) = g(x), so the x-coordinate of that point is the solution.
Finally, turn the cards over.
cut from the notetap to turn over
Seven cards
QuestionHow is the solution of f(x) = g(x) read from the graphs?
AnswerFrom the x-coordinates of the intersection points of y = f(x) and y = g(x).
QuestionIn the graphs, which part gives the solution set of f(x) < g(x)?
AnswerThe x-values where the graph of f is below the graph of g.
QuestionIf the graphs meet at (−1, 2) and (1, 2) and f is above for −1 < x < 1, what is the solution of f ≥ g?
Answer[−1, 1] (equal points together with the region where f is greater).
QuestionWhat solution does the graph give for x/2 + 1 ≤ 0?
AnswerThe graph passes through (−2, 0); x ≤ −2.
QuestionWhat is the solution of −3 ≤ 4 − 7x < 18?
Answer−2 < x ≤ 1.
QuestionWhat is the solution of x² − 4x + 3 < 0?
Answer(x − 1)(x − 3) < 0 gives 1 < x < 3.
QuestionWhat is the solution of x/(x − 2) ≤ 2?
Answerx < 2 or x ≥ 4.
Source: Higher Mathematics 2nd Paper, Classes 11-12 (NCTB), Chapter 1 “Real Numbers and Inequalities”, section 1.8.