krit.club logo

Algebra - Linear Inequalities

Grade 10IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Definition of Inequality: Mathematical statements that compare two expressions using symbols like <, >, ≤, or ≥.

•

Solving Linear Inequalities: Follow the same algebraic steps as equations (addition, subtraction, multiplication, division).

•

The Negative Rule: When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.

•

Number Line Representation: Use an open circle (○) for < and > (exclusive) and a solid circle (●) for ≤ and ≥ (inclusive).

•

Graphical Regions: On a Cartesian plane, use a dashed line for strict inequalities (<, >) and a solid line for non-strict inequalities (≤, ≥). Use a test point like (0,0) to determine which side to shade.

•

Integer Solutions: Often IGCSE questions ask for the 'set of integers' or the 'smallest/largest integer' that satisfies the inequality.

📐Formulae

If a>ba > b, then a+c>b+ca + c > b + c and a−c>b−ca - c > b - c

If ax>bax > b and a>0a > 0, then x>bax > \frac{b}{a}

If ax>bax > b and a<0a < 0, then x<bax < \frac{b}{a} (Sign Reversal)

Compound Inequality: a<x<ba < x < b represents the region where xx is between aa and bb.

💡Examples

Problem 1:

Solve the inequality: 3(x−2)<15+6x3(x - 2) < 15 + 6x

Solution:

3x−6<15+6x3x - 6 < 15 + 6x −3x−6<15-3x - 6 < 15 −3x<21-3x < 21 x>−7x > -7

Explanation:

First, expand the brackets. Then, collect the x-terms on one side. When dividing by -3 to isolate x, the inequality sign is flipped from < to >.

Problem 2:

Find the range of values for xx that satisfies: −5≤2x+1<9-5 \leq 2x + 1 < 9

Solution:

−6≤2x<8-6 \leq 2x < 8 −3≤x<4-3 \leq x < 4

Explanation:

Subtract 1 from all three parts of the inequality. Then, divide all three parts by 2 to isolate x in the middle.

Problem 3:

List the integer values of nn such that −2<n≤3-2 < n \leq 3.

Solution:

{−1,0,1,2,3}\{-1, 0, 1, 2, 3\}

Explanation:

The value -2 is excluded because of the '<' sign (open boundary), while 3 is included because of the '≤' sign (closed boundary).