# How does one construct a 'sign chart' when solving inequalities?

I'm working on solving inequalities for an assignment. The instructions also request that I draw a 'sign chart' along with each solution. I've never heard of a 'sign chart' before, and the internet also seems to have a limited amount of information. From what I can gather...

PurpleMath never actually refers to these diagrams as sign charts either, but I assume these 'factor charts' are synonymous.

The first image is a sign chart for (x + 4)(x – 2)(x – 7) > 0. I know that this has roots at -4, 2, and 7. These are indicated in between columns. The factors are written along rows. The -/+ is determined by evaluating a test point in a range dependent on the columns in each factor. (This is just my understanding - I could by all means be very, very incorrect.) For example - cell "B3" - the range is -4 < TEST < 2. 0 is in this range, so I will evaluate (x-2) for 0. The resulting sign is -, so I mark that in cell B3. I evaluate all other cells in that column, except for the top. The top cell of each column is determined by the resulting sign of all signs below it (in column 3, - * - = +).

However, my intuition fails me when the problems become more complex. Take, for example, 4x^2 + 8x < 6. When solved using the quadratic equation, x equals a messy (-2 +/- sqrt(10) ). I don't understand what to put along the columns and rows. What goes where?

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Consider for example $x-2$ then the interesting point is $x=2$ at the left it will be $-$ at the right $+$ and $0$ for $2$ that's all. For your last inequation consider $s_{\pm}=\frac {-1\pm\sqrt{10}}2$ you may write it in two lines ($x-s_+$ and $x-s_-$) with the same rules (left:$-$, right:$+$) or put it in one line with $3$ columns separated by $s_-$ and $s_+$ : you'll get $+$ outside the roots and $-$ inside (because for large values the sign is the sign of $4$). – Raymond Manzoni Aug 12 '12 at 18:45
It is not clear what your instructor wants a "sign chart" to look like. Simpler would be a number line, with the key points (where function is $0$, or undefined) identified. Then put suitable $\pm$ over each subinterval. – André Nicolas Aug 12 '12 at 18:58
@RaymondManzoni Thanks for the clarification! What if I solve through completing the square? (ex. 4x2 – 2x – 5) What goes along the left side of the rows (I haven't 'factored', so what 'factors' would I use?) – Mark V. Aug 12 '12 at 19:54
@MarkV. : explicitly and if you want to put it in two lines : $x-\frac {-2-\sqrt{10}}2$ (I had a type $-1\to -2$) on one line (or $x-x_1$ with $x_1$ written elsewhere) and $x-\frac {-2+\sqrt{10}}2$ (or $x-x_2$) on the other. – Raymond Manzoni Aug 12 '12 at 20:07

When I hear the phrase 'sign chart,' I do not think of the factor charts that you have included above.

For example, I would do the following for the inequality $(x+2)(1-x) \geq 0$.

1. Find the roots. The roots of this polynomial are $x=-2,1$.
2. Draw a number line, and on this line label my roots.
3. Decide whether the polynomial is positive or negative between every pair of roots.

So here, my sign chart would look like (except not drawn in GIMP):

To determine the $+$ and $-$ signs, we just test any point in that region. When $x = 0$, we have $(2)(1) > 0$. Plugging in $-10$, we have $-10(11) < 0$, and plugging in $10$ we have $(12)(-9)<0$.

If this looks familiar, then that's what you should do. If not, though, then you'll have to ask your teacher/consult your book about what a sign chart is.

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