4 mistakes learners make with inequalities, each one named and explained.
An inequality is solved like an equation, except that multiplying or dividing by a negative turns the sign around.
You shade x > 3 with a solid line, or mark it on a number line with a filled circle.
A line is a line, and an empty circle looks like something left unfinished.
Question. On a number line, how is x > 3 marked at 3?
A common answer. a filled circle
The answer. an open circle
Why. 3 itself is not a solution, and the open circle is what says so.
Look for the line under the sign. No line under it means no line on the graph.
For -2 < n <= 3 you include -2, or leave out the 3.
The two signs look almost identical and the line underneath is the smallest mark in the question.
Question. integers with -2 < n <= 3
A common answer. -2, -1, 0, 1, 2, 3
The answer. -1, 0, 1, 2, 3
Why. The first sign has no line, so -2 itself is not allowed, while the second does, so 3 is.
The line underneath means the number itself is allowed. Mark the ends as filled or open before listing.
4x - 3 < 13 is answered as x > 4, or -3x + 2 >= 11 is answered as x >= -3.
Everybody is told that the sign sometimes flips, and remembering when is harder than remembering that.
Question. -3x + 2 >= 11
A common answer. x >= -3, with the sign left alone
The answer. x <= -3
Why. Dividing by -3 is the moment the sign turns, and nothing else in the question turns it.
Only multiplying or dividing by a negative flips it. Test your answer with one number.
For x > 2 and x < 5 you shade everything above 2 together with everything below 5, which is the whole line.
Both conditions are true statements, so shading both looks like using all the information.
Question. Which numbers satisfy both x > 2 and x < 5?
A common answer. every number, since each part is satisfied somewhere
The answer. only the numbers between 2 and 5
Why. And means both at once, so only the overlap counts.
Shade each one lightly, then keep only the part shaded twice.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.