Algebra, Year 8: brackets, inequalities, lines and indices

x = 4

提示You could divide both sides by 3 first.

为什么Dividing by 3 gives x + 2 = 6, so x = 4. Expanding gives 3x + 6 = 18 and the same answer. Check: 3 × (4 + 2) = 18.

x = 5

提示What must the bracket be worth?

为什么Five lots of the bracket make 20, so the bracket is 4. Then x − 1 = 4 gives x = 5. Check: 5 × (5 − 1) = 20.

x = 3

提示Halve both sides, then treat it as a two-step equation.

为什么Dividing by 2 gives 3x + 1 = 10, so 3x = 9 and x = 3. Check: 2 × (9 + 1) = 20.

x = −0.5

提示Dividing first gives a decimal; expanding first works just as well.

为什么Expanding: 4x + 12 = 10, so 4x = −2 and x = −0.5. Dividing first: x + 3 = 2.5. When the division is awkward, expanding is often the easier route.

An open (hollow) circle

提示It is the opposite of the dot used when the end value counts.

为什么An unfilled circle is used for < and >, where the end value is left out. A filled circle is used for ≤ and ≥, where it counts.

−1, 0, 1, 2

提示Check each end: one is included and one is not.

为什么x can equal −1, because the sign is ≤, but it must be less than 3, so 3 is left out. Zero is an integer and must not be forgotten.

x ≤ 4

提示Filled means the end value counts.

为什么The arrow covers every number below 4, and the filled circle says that 4 itself is included, so the sign is 'less than or equal to'.

−2 < x ≤ 5

提示Write x in the middle and decide each sign separately.

为什么The open circle at −2 means x is greater than −2 but not equal to it. The filled circle at 5 means x can be 5.

−3

提示Twice a negative number is negative; then add one.

为什么2 × (−2) = −4, and −4 + 1 = −3. Tables of values for straight lines should always include some negative x-values.

y = −2, 1, 4

提示Each y is 3 more than the one before.

为什么3 × 0 − 2 = −2, 3 × 1 − 2 = 1 and 3 × 2 − 2 = 4. Equal steps in x give equal steps in y, which is why the points lie on a straight line.

Re-check the calculation for that point

提示A linear rule cannot make a kink.

为什么Every point of y = 2x + 1 lies on one straight line, so a stray point signals an arithmetic slip, often with a negative number.

x = −2

提示Ask which coordinate never changes.

为什么Every point on the line has an x-coordinate of −2, whatever its height. The equation states the one thing that is always true.

Horizontal

提示Every point on it is the same height above the x-axis.

为什么All its points have a y-coordinate of 2, such as (0, 2) and (5, 2), so the line runs level, two units up.

y = 0

提示Think about what all its points have in common.

为什么Every point on the x-axis has a height of nothing: (1, 0), (2, 0), (−4, 0). In the same way, the y-axis is x = 0.

(4, −1)

提示One line fixes the across number; the other fixes the up-down number.

为什么The crossing point lies on both lines, so its x-coordinate is 4 and its y-coordinate is −1.

a⁷

提示Write each power out as a string of a's and count them.

为什么a³ is a × a × a and a⁴ is a × a × a × a. Together that is seven a's multiplied, so the indices are added.

a⁶

提示Cancel two a's from the top.

为什么Eight a's on top and two underneath: two pairs cancel, leaving six. When dividing powers of one letter, subtract the indices.

6a⁵

提示Multiply the numbers; deal with the letters separately.

为什么2 × 3 = 6 and a² × a³ = a⁵. The numbers are multiplied in the ordinary way; only the indices are added.

No — the bases are different

提示Write both out in full and see what you get.

为什么a × a × a × b × b cannot be written as a single power, because the letters differ. It stays as a³b².