2.4.2 Linear Inequalities
2.4.2 Linear Inequalities
Linear Inequality
An inequality in which the highest power of variable/variables is 1 is known as a linear inequality.
e.g.
\( 2x+3≤9, 5<10x≤15 \)
To solve linear inequalities
- Apply the same rules as when solving linear equations
- Remember to keep the inequality sign throughout
- When multiplying or dividing both sides by a negative number, the sign of the inequality must be flipped
- Never multiply or divide by a variable that can be either positive or negative
- The most secure way to rearrange is to simply add and subtract to move all the terms to one side
Worked example:
Find the integer values that satisfy the inequality \( 2<2x≤10 \).
Divide all the three terms by \( 2 \)
\( \frac{2}{2}< \frac{2x}{2} \leq \frac{10}{2} \)
\( 1<x≤5 \)
List all the integers in the interval
\( 2,3,4,5 \)
\( 2,3,4,5 \)
Linear Inequalities on a number line
In order to represent inequalities on a number line
- Determine whether an open or closed circle is required
- An open circle indicates that the value is not included
- A closed circle indicates that the value is included
- Indicate the solution set by drawing a straight line to the left or right side of the number, or by drawing a straight line between the circles
Worked example:
Represent \( -3<x≤6 \) on a number line.
An open circle should be indicated above \( -3 \) and a closed circle above \( 6 \)
Draw a line between the circles to indicate any value that exists between them
Test yourself
Question 1:
Find the integer values of n that satisfy the inequality
\( 18-2n<6n≤30+n \)
[3]
Question 2:
Solve the inequality
\( 3(x-2)<7(x+2)\)
[3]
Question 3:
Solve the inequality
\( 3m+12≤8m-5 \)
[2]
Question 4:
Find the integer value of n that satisfy the inequality
\( 15≤4n<28 \)
[3]
Question 5:
Solve the inequality
\( n+7<5n-8 \)
[3]