The equation x = 7 has one point for an answer. The inequality x < 7 has a whole stretch, every number left of 7. Solving one is almost like solving an equation, except at one place where the sign turns round.
1. What an inequality is
An inequality in one variable says one expression is less than, greater than, at most or at least another: x < 7, x > 7, x ≤ 7, x ≥ 7. A linear equation has a single solution, a linear inequality has many.
a ≤ b and a ≥ b are weak inequalities, and they obey the same basic axioms. The absolute value of any real number is greater than or equal to zero, which is one of them. a < x < b is a compound inequality, because it says two things: a < x and x < b.
An inequality true only for some values of the variable is conditional: x + 3 > 3 holds only when x > 0. One true for every value is unconditional: x + 1 > x.
pick the kindchip, then bin
Conditional or unconditional?
Pick up a chip, then drop it in a box. True for every x means unconditional.
Right. Next one.
Not yet. Try x = −1 or x = 0: is it still true?
All right. x + 3 > 3 and x > 7 hold for some values; x + 1 > x, x² ≥ 0 and |x| ≥ 0 hold for all.
2. ax > b, and when the sign turns
In ax > b with a > 0, divide both sides by a: x > b/a, the interval (b/a, ∞). With a < 0 the solution is (−∞, b/a). With a = 0 the inequality reads 0 > b: no solution when b ≥ 0, every real x when b < 0. The book says a = 0 never has a solution; the correct rule is that only b ≥ 0 leaves none.
change a and bdrag the handles
Which way does ax > b point?
That is the rule: multiply or divide an inequality by a negative number, or turn both sides over, and the sign reverses. If a > b then −a < −b, and for positive a, b, 1/a < 1/b. For example x < −2 times (−1) gives −x > 2. The book says a > b ⇒ 1/a < 1/b for all a, b; it holds only when a and b have the same sign. Take a = 2, b = −1 and you get 1/2 > −1.
3. Solving, then drawing it
In 3x − 2 > 2x − 1, add −2x to both sides: x − 2 > −1, so x > 1. From x ≤ (1/2)x + 1 we get (1/2)x ≤ 1, so x ≤ 2; the solution set is S = {x ∈ ℝ : x ≤ 2}, drawn as a filled dot at 2 with an arrow to the left.
Write 12 ≥ 3 − 3x as 3 − 3x ≤ 12, then add −3 to get −3x ≤ 9. Multiplying by −1 now reverses it to 3x ≥ −9, so x ≥ −3. A compound inequality is worked on both sides at once: −3 ≤ 4 − 7x < 18 gives −2 < x ≤ 1, because dividing by −7 reverses both signs. And −3 ≤ 2x < 8 has the integer solutions −1, 0, 1, 2, 3, five of them.
step by steppick one
Solve it, see it on the line
Whether the dot is filled or empty, the sign tells you: with ≤ or ≥ the end point is in the solution, a filled dot; with < or > it is not, an empty dot. Now put each answer with its picture.
dot and arrowchip, then bin
How does it look on the line?
Pick up a chip, then drop it in a box. < or > gives an empty dot.
Right. Next one.
Not yet. First see whether the sign includes equals, then whether x is bigger or smaller.
All in place. For −2 < x ≤ 1, an empty dot at −2, a filled dot at 1 and the line between.
4. Check yourself
Three questions in the exam's style. The first is about a definition.
check yourselfpick one answer
Question 1
a < x < b is called:
an equation does not fit here; the right answer is a compound inequality. It states two things, a < x and x < b, so it is a compound inequality.
a weak inequality does not fit here; the right answer is a compound inequality. It states two things, a < x and x < b, so it is a compound inequality.
Right. It states two things, a < x and x < b, so it is a compound inequality.
an unconditional inequality does not fit here; the right answer is a compound inequality. It states two things, a < x and x < b, so it is a compound inequality.
In the second, think of every x.
check yourselfpick one answer
Question 2
Which of the following is an unconditional inequality?
Right. x + 1 > x is true for every value of the variable, so it is unconditional.
x + 3 > 3 does not fit here; the right answer is x + 1 > x. x + 1 > x is true for every value of the variable, so it is unconditional.
x > 7 does not fit here; the right answer is x + 1 > x. x + 1 > x is true for every value of the variable, so it is unconditional.
x² > 4 does not fit here; the right answer is x + 1 > x. x + 1 > x is true for every value of the variable, so it is unconditional.
In the last, bring x to one side first.
check yourselfpick one answer
Question 3
Which is the solution set of 5x − 3 ≤ 3x + 5?
(4, ∞) does not fit here; the right answer is (−∞, 4]. 2x ≤ 8 ⇒ x ≤ 4, so (−∞, 4].
[4, ∞] does not fit here; the right answer is (−∞, 4]. 2x ≤ 8 ⇒ x ≤ 4, so (−∞, 4].
(−∞, 4) does not fit here; the right answer is (−∞, 4]. 2x ≤ 8 ⇒ x ≤ 4, so (−∞, 4].
Right. 2x ≤ 8 ⇒ x ≤ 4, so (−∞, 4].
The main points of the note, on cards.
cut from the notetap to turn over
Eight cards
QuestionHow do the numbers of solutions of a linear equation and a linear inequality differ?
AnswerThe equation has exactly one solution (root); the inequality has many.
QuestionFor ax > b with a > 0, what is the solution?
Answerx > b/a, i.e. the interval (b/a, ∞).
QuestionFor ax > b with a < 0, in which interval does the solution lie?
AnswerIn (−∞, b/a).
QuestionFor ax > b with a = 0 and b < 0, what is the solution?
Answer0 > b is true, so every real x is a solution (for b ≥ 0 there is none).
QuestionWhat does x < −2 become when multiplied by (−1)?
Answer−x > 2 (multiplying by a negative reverses the sign).
QuestionWhat kind of inequality is x + 1 > x and why?
AnswerUnconditional, because it is true for every value of x.
QuestionWhy is −1 < x < 5 called a compound inequality?
AnswerBecause it states two things: −1 < x and x < 5.
QuestionWhat is the solution of −3 ≤ 4 − 7x < 18?
Answer−2 < x ≤ 1.
Source: Higher Mathematics 2nd Paper, Classes 11-12 (NCTB), Chapter 1 “Real Numbers and Inequalities”, section 1.5-1.6.