Every card in Algebra, Year 8: brackets, inequalities, lines and indices

x = 4

HintYou could divide both sides by 3 first.

WhyDividing 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

HintWhat must the bracket be worth?

WhyFive 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

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

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

x = −0.5

HintDividing first gives a decimal; expanding first works just as well.

WhyExpanding: 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

HintIt is the opposite of the dot used when the end value counts.

WhyAn 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

HintCheck each end: one is included and one is not.

Whyx 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

HintFilled means the end value counts.

WhyThe 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

HintWrite x in the middle and decide each sign separately.

WhyThe 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

HintTwice a negative number is negative; then add one.

Why2 × (−2) = −4, and −4 + 1 = −3. Tables of values for straight lines should always include some negative x-values.

y = −2, 1, 4

HintEach y is 3 more than the one before.

Why3 × 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

HintA linear rule cannot make a kink.

WhyEvery 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

HintAsk which coordinate never changes.

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

Horizontal

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

WhyAll 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

HintThink about what all its points have in common.

WhyEvery 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)

HintOne line fixes the across number; the other fixes the up-down number.

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

a⁷

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

Whya³ is a × a × a and a⁴ is a × a × a × a. Together that is seven a's multiplied, so the indices are added.

a⁶

HintCancel two a's from the top.

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

6a⁵

HintMultiply the numbers; deal with the letters separately.

Why2 × 3 = 6 and a² × a³ = a⁵. The numbers are multiplied in the ordinary way; only the indices are added.

No — the bases are different

HintWrite both out in full and see what you get.

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