What is the solution of 5x − 9 −1 Choose the right answer

A. x < −1. 5x < −14 − (−9), so 5x < 5 × −1; divide by 5: x < −1.

The least whole-number solution of 2x + 6 > −5 is □

−5. 2x > −11, so x > −11/2. That isn't whole, and the least whole number above it is −5.

Solve −13 − 3x > 5 without dividing by a negative: x terms to the right, numbers to the left, giving −18 > □x, so x < □ Fill in each blank

3, −6. Add 3x and take away 5 on both sides: −13 − 5 > 3x, that is −18 > 3x; divide by the positive 3: −6 > x, so x < −6.

Linear inequalities in one unknown · Ages 14–15

Solving linear inequalities

Solve as an equation, turning the sign when dividing by a negative; check one solution.

How to do it

Method

Solve as an equation, turning the sign when dividing by a negative; check one solution

5 − 2x ≤ 11: −2x ≤ 6, x ≥ −3

Another way

x to the positive side

−6 ≤ 2x, x ≥ −3

In full

Solve it as an equation is solved: open brackets, clear fractions, bring the x terms to one side and the numbers to the other, then divide by x's coefficient. The one difference: multiplying or dividing by a negative turns the sign round. 3x − 5 > 7: 3x > 12, x > 4. 5 − 2x ≤ 11: −2x ≤ 6, and dividing by −2 turns it, x ≥ −3. Or bring the x terms to the side where their coefficient is positive, and you never divide by a negative: 5 − 2x ≤ 11; add 2x and take away 11: −6 ≤ 2x, so x ≥ −3. Then check with one solution, say x = 0: 5 − 0 = 5 ≤ 11, true. Whole-number solutions: the least for x > 4 is 5; the greatest for x ≤ 2.5 is 2; the least for x ≥ −3 is −3 itself.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
What is the solution of 5x − 9 < −14?
- A. x < −1
- B. x < 1
- C. x ≤ −1
- D. x > −1
Choose the right answer
Show the answer and working
Answer: A. x < −1
5x < −14 − (−9), so 5x < 5 × −1; divide by 5: x < −1.
Example 2
The least whole-number solution of 2x + 6 > −5 is
Show the answer and working
Answer: −5
2x > −11, so x > −112. That isn't whole, and the least whole number above it is −5.
Example 3
Solve −13 − 3x > 5 without dividing by a negative: x terms to the right, numbers to the left, giving −18 > x, so x <
Fill in each blank
Show the answer and working
Answer: 3, −6
Add 3x and take away 5 on both sides: −13 − 5 > 3x, that is −18 > 3x; divide by the positive 3: −6 > x, so x < −6.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Solve 4x + 6 ≥ −14: x ≥
Answer:
2.Solve 8(x − 4) ≤ 5x − 29: x ≤
Answer:
3.x = −3 is a solution of −2 − 6x ≥ 18.
True or false?
Answer:
4.The greatest whole-number solution of x − 53 < 4 is
Answer:
5.Solve 1 − 9x > 10 without dividing by a negative: x terms to the right, numbers to the left, giving −9 > x, so x <
Fill in each blank
Answer:

All the answers

1. −5
2. 1
3. False
4. 16
5. 9, −1

Free worksheet: Solving linear inequalitiesWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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