Find the missing number by undoing one operation at a time. Then explore which numbers make an inequality true.
An equation says two expressions have the same value. In 3x + 4 = 19, x stands for a number we need to find: three times that number, plus 4, equals 19. Adding or subtracting the same amount on both sides keeps them equal. So does multiplying or dividing both sides by the same nonzero number.
For these two-step equations, undo the operations in reverse order. In 3x + 4 = 19, x was multiplied by 3 and then 4 was added. Subtract 4 from both sides to get 3x = 15. Then divide both sides by 3 to get x = 5.
An inequality compares two values: < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to. When you multiply or divide both sides by a negative number, reverse the inequality symbol: < becomes >, and ≤ becomes ≥ (and the other way around). For example, 2 < 3, but multiplying both sides by −1 gives −2 > −3. On a number line, a filled dot includes that number; an open dot leaves it out.
A triangle’s three angles add to 180°. Complementary angles add to 90°. Supplementary angles add to 180°. Write that as an equation, then undo it.
Try it. Then open the steps.
Example 1 of 5
School store, a pen on the counter
Three identical notebooks and a $4 pen cost $19. How much is one notebook?
Symbolic relationship · not drawn to scale
n + n + n + 4 = 19
Answer: $5
Example 2 of 5
Solve 2x − 5 ≥ 9. What is the smallest integer x that works? (Integers are whole numbers and their negatives, like −2, −1, 0, 1, 2.)
Answer: 7
Example 3 of 5
Solve −3x + 4 < 19.
Answer: x > −5
Example 4 of 5
Ice cream, the bill on the counter
Three identical scoops and a $2 topping cost $11. How much is one scoop?
Answer: $3
Example 5 of 5
Both sides stay balanced
2x + 5 = 17. What is x?
Answer: 6
Easy to miss: Changing only part of one side. For example, 3x + 7 = 19 does not become 3x = 19 + 7.
Do this: Subtract 7 from both sides: 3x = 19 − 7 = 12. Then divide both sides by 3 to find x = 4.
Easy to miss: Forgetting to flip the inequality when dividing by a negative.
Do this: Dividing −2x > 8 by −2 flips it to x < −4.
31 problems · Camp, Trail, Summit
