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

x = 4

HintYou could divide both sides by 3 first.

The 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?

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

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