Linear Inequalities — Challenge

Solve it like an equation, with one extra rule: multiplying or dividing by a negative reverses the sign.

What these problems cover

These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

What is the least integer x for which 3x-5>7?

  1. 3
  2. 4
  3. 5
  4. 6
  5. 12
Worked answer

5

Add 5 to both sides: 3x > 12.

Divide by 3: x > 4.

The inequality is strict, so 4 does not count and the least integer is 5.

Problem 2

What is the greatest integer x for which -2x≥ 8?

  1. -6
  2. -5
  3. -4
  4. 4
  5. -2
Worked answer

-4

Divide both sides by -2, which reverses the inequality: x≤ -4.

Equality is allowed, so -4 itself qualifies.

Among integers at most -4, the greatest is -4 itself.

Problem 3

What is the least integer x for which 4-x<1?

  1. 3
  2. 4
  3. 5
  4. 6
  5. -3
Worked answer

4

Subtract 4 from both sides: -x < -3.

Multiply by -1 and reverse the inequality: x > 3.

The least integer greater than 3 is 4. Check: 4-4 = 0, which is less than 1.

Problem 4

What is the greatest integer x for which 5x≤ -15?

  1. -5
  2. -4
  3. -3
  4. 3
  5. -1
Worked answer

-3

Divide both sides by 5, which is positive, so the inequality does not reverse: x≤ -3.

Equality is allowed, so -3 qualifies and is the greatest such integer.

Problem 5

How many integers x satisfy -1<2x+3≤ 9?

  1. 4
  2. 5
  3. 6
  4. 7
  5. 8
Worked answer

5

Subtract 3 from all three parts: -4 < 2x≤ 6.

Divide all three parts by 2: -2 < x≤ 3.

The left end is excluded and the right included, so the integers are -1,0,1,2,3, which is 5 of them.

Problem 6

What is the greatest integer x for which 2(x+1)≥ 3x-4?

  1. 2
  2. 4
  3. 5
  4. 6
  5. 7
Worked answer

6

Expand the left side: 2x+2≥ 3x-4.

Subtract 2x and add 4 to both sides: 6≥ x, that is x≤ 6.

Equality is allowed, so 6 qualifies. Check: 2(7) = 14 and 3(6)-4 = 14.

Problem 7

Which of these inequalities has the solution x<2?

  1. 3x<6
  2. 3x>6
  3. x-2>0
  4. x+2<0
  5. 2x>4
Worked answer

3x<6

Solve each one and compare. Dividing 3x<6 by 3 gives x<2, which matches.

3x>6 and 2x>4 both give x>2, the opposite direction.

x-2>0 gives x>2, and x+2<0 gives x<-2, a different set.

Problem 8

What is the least integer x for which x2>5?

  1. 6
  2. 10
  3. 2.5
  4. 11
  5. 20
Worked answer

11

Multiply both sides by 2: x > 10.

The inequality is strict, so 10 does not count.

The least integer greater than 10 is 11.

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