Linear inequalities
Solve linear inequalities, including those requiring reversal of the inequality sign, and represent solutions on a number line.
Worked examples
Solving a one-step inequality
Straightforward
Problem
Solve and state the boundary value of .
1
Subtract 5 from both sides to isolate .
2
Simplify.
Answer
The boundary value of is . The solution is .
Solving a two-step inequality with sign reversal
Moderate
Problem
Solve and state the boundary value of .
1
Subtract 5 from both sides.
2
Divide both sides by . Because you are dividing by a negative number, the inequality sign reverses.
Answer
The boundary value of is . The solution is .
Solving an inequality with brackets and variables on both sides
Challenging
Problem
Solve and state the boundary value of .
1
Expand the brackets on the left-hand side.
2
Subtract from both sides to collect terms on the left.
3
Subtract 6 from both sides.
4
Divide both sides by . Reverse the inequality sign because you are dividing by a negative number.
Answer
The boundary value of is . The solution is .
Practise
Q1·Straightforward
Solve . What is the boundary value of ?
Explanation
Dividing both sides by 2 gives . The boundary value is .
Q2·Straightforward
Solve . What is the boundary value of ?
Explanation
Subtracting 5 from both sides gives . The boundary value is .
Q3·Straightforward
Solve . What is the boundary value of ?
Explanation
Adding 4 to both sides gives . The boundary value is .
Q4·Straightforward
Solve . What is the boundary value of ?
Explanation
Multiplying both sides by 3 gives . The boundary value is .
Q5·Moderate
Solve . What is the boundary value of ?
Explanation
Subtract 3: . Divide by 2: . The boundary value is .
Q6·Moderate
Solve . What is the boundary value of ?
Explanation
Add 7: . Divide by 3: . The boundary value is .
Q7·Moderate
Solve . What is the boundary value of ?
Explanation
Divide both sides by and reverse the sign: . The boundary value is .
Q8·Moderate
Solve . What is the boundary value of ?
Explanation
Subtract 5: . Divide by and reverse: . The boundary value is .
Q9·Challenging
Solve . What is the boundary value of ?
Explanation
Subtract : . Subtract 1: . Divide by 2: . The boundary value is .
Q10·Challenging
Solve . What is the boundary value of ?
Explanation
Subtract : . Add 3: . Divide by 3: . The boundary value is .
Q11·Challenging
Solve . What is the boundary value of ?
Explanation
Expand: . Subtract : . Subtract 8: . The boundary value is .
Q12·Challenging
Solve . What is the boundary value of ?
Explanation
Expand: . Subtract : . Subtract 6: . Divide by and reverse: . The boundary value is .