Solving and graphing

A step-by-step method for any linear inequality:

• Clear fractions by multiplying both sides by the LCM of the denominators. The LCM is positive, so the sign stays.
• Expand brackets.
• Collect the x terms on one side and the numbers on the other.
• Divide by the number in front of x, reversing the sign if it is negative.

Example: (x + 1)/2 ≥ (2x – 3)/3.

• Multiply by 6: 3(x + 1) ≥ 2(2x – 3), so 3x + 3 ≥ 4x – 6.
• Collect: 3x – 4x ≥ –6 – 3, so –x ≥ –9.
• Divide by –1 and reverse: x ≤ 9.

Example: 2(x + 3) – 5 < 3(x – 1). Expand: 2x + 1 < 3x – 3. Collect: –x < –4. Reverse: x > 4.

Graphing on a number line: mark the end point with an open circle for < or > (the point is not included) and a filled circle for ≤ or ≥ (it is included), then shade the side that works. x ≤ 9 is a filled circle at 9 with everything to the left shaded; x > 4 is an open circle at 4 with everything to the right shaded.

Word problems. The key is turning words into signs:

Averages: a student scored 68 and 79 in two tests and wants an average of at least 75 over three. If the third mark is x, then (68 + 79 + x)/3 ≥ 75, so 147 + x ≥ 225 and x ≥ 78. The least mark needed is 78.

Two conditions together: find pairs of consecutive odd numbers, both less than 12, whose sum is more than 15. Call them x and x + 2.

• x + 2 < 12 gives x < 10, and x + (x + 2) > 15 gives x > 6.5.
• So x is an odd number with 6.5 < x < 10: x = 7 or 9. The pairs are (7, 9) and (9, 11).

Shapes: the longest side of a triangle is twice the shortest, and the third side is 4 cm longer than the shortest. If the perimeter is at least 44 cm and the shortest side is x, then x + 2x + (x + 4) ≥ 44, so 4x ≥ 40 and x ≥ 10 cm.