Skip to main content

Back to the full dot-point answer

NSWMaths AdvancedQuick questions

Year 11: Functions

Quick questions on Intervals, inequalities and absolute value for HSC Maths Advanced: interval notation and number-line graphs, solving linear inequalities (and when to reverse the sign), solving quadratic inequalities from the parabola, and the absolute value definition with |x| < k and |x| > k

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

Why does the sign flip?
Show answer
Multiplying by a negative reflects every number through 00 on the number line, which reverses their order: 2<52 < 5 is true, but multiplying by −1-1 gives −2-2 and −5-5, and −2>−5-2 > -5. The order has turned around, so the inequality symbol must turn around with it.
What are solving linear inequalities?
Show answer
A linear inequality is solved almost exactly like a linear equation: do the same thing to both sides to keep it balanced. You may add or subtract any number from both sides freely, and you may multiply or divide both sides by any positive number with no change. There is exactly one extra rule, and it is the whole difficulty of the topic.
What is solving quadratic inequalities from the parabola?
Show answer
A quadratic inequality like x2−x−6>0x^2 - x - 6 > 0 asks "for which xx is this quadratic positive (or negative)?" You do not solve it like an equation and you do not simply divide. The reliable method is to look at the graph: a quadratic is a parabola, and the sign of the quadratic is just whether the parabola is above or below the xx-axis.
What is read the sign off the parabola, stage by stage?
Show answer
The cleanest way to see a quadratic inequality is to sketch the parabola and shade where it sits on the wanted side of the axis. Below, x2−x−6>0x^2 - x - 6 > 0 is built up: axis and intercepts, then the curve, then the sign of each region, then the solution read onto a number line. The answer is x<−2x < -2 or x>3x > 3.
What is absolute value as distance?
Show answer
The absolute value ∣x∣|x| is most usefully defined as the distance of xx from 00 on the number line. So ∣5∣=5|5| = 5 and ∣−5∣=5|{-5}| = 5: both are 55 units from the origin. Distance is never negative, so ∣x∣≥0|x| \ge 0 for every real xx, and ∣−x∣=∣x∣|{-x}| = |x|. There are two equivalent ways to write it down:

Have a question we have not covered?

This dot-point answer is short enough that we have not extracted many short questions yet. Read the full dot-point answer or ask Mo, our study assistant, in the chat for follow ups.

All Maths AdvancedQ&A pages