Linear Inequalities

Subtract the same number from both sides.

1. Subtract 5 from both sides of the inequality.
2. The solution is x<4.

Divide both sides by the positive coefficient of the unknown.

1. Divide both sides by 3.
2. Because 3 is positive, the inequality keeps its direction, giving x\ge5.

When dividing by a negative number, reverse the inequality sign.

1. Subtract 12 from both sides: -2x>-6.
2. Divide by -2 and reverse the sign, giving x<3.

Undo the subtraction before undoing the multiplication.

1. Add 7 to each side to obtain 4x\le16.
2. Divide by 4: x\le4. The value 4 is included.

Collect the unknown terms on one side and the constants on the other.

1. Subtract 2x and then 2 from each side to obtain 3x>15.
2. Divide by 3, giving x>5.

You can divide by the number outside the bracket before removing the bracket.

1. Divide by 2: x-3\le5.
2. Add 3, giving x\le8.

A filled circle means that number is included; an empty circle means it is not.

1. The line runs from -2 to 3, so x is between -2 and 3.
2. The filled circle at -2 means -2 is included: -2\le x.
3. The empty circle at 3 means 3 is not included: x<3.
4. Together: -2\le x<3.

Multiply both sides by the positive denominator.

1. Multiply by 3 to get x+2>12.
2. Subtract 2, giving x>10.

Apply each operation to all three parts of the inequality.

1. Subtract 1 throughout: -3<3x\le9.
2. Divide all three parts by 3.
3. The solution is -1<x\le3; -1 is not included and 3 is included.

Subtract 3 from all three parts, then divide all three parts by 2.

1. Subtract 3 from all three parts: -7\le2n<6.
2. Divide all three parts by 2: -3.5\le n<3.
3. n is an integer, so the least value is -3. 3 is not included, so the greatest value is 2.
4. The possible values are -3,-2,-1,0,1,2.

Expand the bracket carefully, keeping the negative coefficient of the unknown.

1. Expand to get 8-4x>3x-6.
2. Add 4x and 6 to both sides, giving 14>7x.
3. Divide by 7: x<2.

Multiply both sides by the lowest common multiple of the denominators.

1. Multiply by 6, a positive number, to obtain 2(2x-1)\le3(x+4).
2. Expand: 4x-2\le3x+12.
3. Subtract 3x and add 2, giving x\le14.

Solve the inequality first, then decide which integers it allows.

1. Subtract 2x and subtract 3 from both sides: 3x>-12.
2. Divide by 3: x>-4.
3. -4 itself is not allowed, so the least integer is -3.

Multiply both sides by 3 first, then solve as usual.

1. Multiply both sides by 3: 4y+9<45.
2. Subtract 9: 4y<36. Divide by 4: y<9.
3. 9 itself is not allowed, so the greatest integer is 8.

Write an inequality including the one booking charge before finding the whole-number maximum.

1. For n tickets, the total cost is 3+4.5n, so 3+4.5n\le30.
2. Subtract 3 and divide by 4.5 to obtain n\le6.
3. The greatest possible whole number is 6 tickets.

Solve the two inequalities separately, then take the overlap.

1. The first inequality expands to 6x-2>x+8, giving 5x>10 and hence x>2.
2. The second inequality gives -x\ge-4, so reversing the sign on division by -1 gives x\le4.
3. Both conditions hold when 2<x\le4.

Do the same thing to all three parts: first multiply by 2.

1. Multiply all three parts by 2: 10<3x+1\le22.
2. Subtract 1 from all three parts: 9<3x\le21.
3. Divide all three parts by 3: 3<x\le7.

Solve the combined inequality first, then list only its integer solutions.

1. Multiply all three parts by 4: -8\le3x-5<7.
2. Add 5 to all three parts: -3\le3x<12.
3. Divide all three parts by 3: -1\le x<4.
4. -1 is included and 4 is not, so the integers are -1,0,1,2,3.

Solve each inequality on its own, then list the integers that satisfy both.

1. First inequality: expand to get 3x-6<x+9, so 2x<15 and x<7.5.
2. Second inequality: add 2x and subtract 1 from both sides to get 10\le5x, so x\ge2.
3. The integers with x\ge2 and x<7.5 are 2,3,4,5,6,7.

Write an inequality for the total mass, including the technician, then solve it for n.

1. The total mass is 78+24.5n kg, so 78+24.5n\le630.
2. Subtract 78: 24.5n\le552. Divide by 24.5: n\le22.53\ldots
3. n is a whole number of boxes, so the greatest value is 22. Check: 78+24.5\times22=617 kg, but 78+24.5\times23=641.5 kg is too much.