Linear Inequations
ICSE Class 8 · Mathematics · Chapter 15
What this chapter covers
- Inequalities in everyday life
- Linear inequations in one variable
- Replacement set and solution set
- Properties of inequalities
- Solving linear inequations
- Representing solutions on the number line
- Word problems on linear inequations
Key points
- An inequality compares two quantities using <, >, ≤ or ≥.
- "At least" means ≥; "at most" and "not more than" mean ≤.
- A linear inequation in one variable has one variable, with highest power 1.
- The replacement set lists the values the variable may take; the solution set is the part of it that makes the inequation true.
- Adding or subtracting the same number on both sides keeps the sign.
- Multiplying or dividing both sides by a positive number keeps the sign.
- Multiplying or dividing both sides by a negative number reverses the sign.
- Swapping the two sides reverses the sign: 5 ≤ x is the same as x ≥ 5.
- After solving, always choose the answer from the replacement set and test the boundary value.
- On a number line, filled dots show integer solutions; for any-value solutions, a filled circle includes the end point and a hollow circle leaves it out.
Questions answered
Write as an inequality: "The marks m needed to pass are at least 33."
m ≥ 33.
Which of these is a linear inequation in one variable: (a) 5x - 2 > 8, (b) y2 ≤ 9, (c) a + b < 4?
Only (a). In (b) the power of y is 2, and (c) has two variables.
What is the difference between a replacement set and a solution set?
The replacement set is the set of values the variable may take. The solution set is the part of it that makes the inequation true.
Find the solution set of x + 3 ≤ 6 when x in W.
x ≤ 3, so the solution set is \0, 1, 2, 3\.
Why does the sign reverse when you divide both sides of an inequality by a negative number? Explain with an example.
Dividing by a negative number is the same as dividing by -1 and then by a positive number. Dividing by -1 reflects the numbers across 0 on the number line, so their order turns round, and dividing by a positive number keeps that new order. For example, 2 < 6, but 2 ÷ (-2) = -1 and 6 ÷ (-2) = -3, and -1 > -3.
Solve 5x - 4 > 11, where x in N and x < 7.
5x > 15, so x > 3. The solution set is \4, 5, 6\.
Solve 10 - 2x ≥ 4, where x in Z and x ≥ -1.
-2x ≥ -6, so x ≤ 3 (sign reversed). The solution set is \-1, 0, 1, 2, 3\.
Describe how you would show x > -1 on a number line when x can be any number.
Draw a hollow circle at -1, because -1 is not included, and a thick line running from it to the right, ending in an arrow.