☰ Year plan · Mathematics

Equations and inequalities

Week 8 · Solving equations and exploring and solving inequalities · 3 lessons
3

Solving inequalities and checking the solutions

Discover Mathematics P6: Term 1, Unit 4, Lessons 1–3
GoalSolve one-step inequalities, test values and graph the solutions.
New words
solve · find all the values that make the statement truetest a value · put a number into the inequality to see if it is trueboundary · the number where the solutions start or stop
Explanation

We solve an inequality like an equation, with inverse operations on both sides. For x + 3 > 7, subtract 3 from both sides: x > 4. For 2x ≤ 10, divide both sides by 2: x ≤ 5. For x − 2 ≥ 6, add 2: x ≥ 8. Then check: test a value inside the solution set (it should make the inequality true) and the boundary value (it works only for ≤ and ≥). Graph the answer with an open or filled circle and an arrow. Real limits give inequalities too: a lift carries at most 600 kg and a person with luggage weighs 85 kg, so the extra load x must satisfy x + 85 ≤ 600, which gives x ≤ 515.

★Remember
Solve with inverse operations, test a value, test the boundary, then graph.
Worked examples
012345678910
Solve x + 3 > 7. Subtract 3: x > 4. Test: 3 gives 6 > 7 (false), 4 gives 7 > 7 (false), 5 gives 8 > 7 (true). The circle at 4 is open.
012345678910
Solve 2x ≤ 10. Divide by 2: x ≤ 5. Test: 5 gives 10 ≤ 10 (true), 6 gives 12 ≤ 10 (false). The circle at 5 is filled.
500505510515520525530
The lift story: x + 85 ≤ 600 gives x ≤ 515. The extra load can be up to and including 515 kg.
Class activity

Whole-number detectives. A grown-up writes an inequality such as x + 2 < 9. List all the whole numbers from 0 to 12 that are solutions, then draw the answer on a number line. Swap: you write the inequality, and the grown-up lists the solutions. Check with a test value.

Practice
1
Solve x − 2 ≥ 6.
2
Solve 3x < 12.
3
List the whole-number solutions of x + 1 ≤ 4.
4
Describe the graph of x ≥ 6.
5
Is 7 a solution of x ÷ 2 > 3? Solve to check.