Linear Inequalities: common misconceptions

If your child says… “At most four means less than four.”

Ask together

Does exactly four exceed the limit?

What’s true

‘At most’ and ‘at least’ include equality; ‘less than’ and ‘greater than’ exclude it.

If your child says… “The squared one is linear because it compares amounts.”

Ask together

What power does x have in the last row?

What’s true

A nonzero squared term prevents an inequality from being linear.

If your child says… “Dividing by positive two reverses the sign.”

Ask together

When both sides are halved, does the smaller side become larger?

What’s true

Positive division preserves order.

If your child says… “The answer includes fractions even when x must be an integer.”

Ask together

Is one and a half an integer, even though it lies below the bound?

What’s true

Keep only solutions belonging to the allowed number set.

If your child says… “The endpoint is always left out.”

Ask together

Does one meet ‘at least one’?

What’s true

An endpoint belongs when the comparison allows equality there.

If your child says… “The total only needs to reach three.”

Ask together

If the total were three, what would dividing it by two give?

What’s true

The required average applies after dividing the total by the number of tests.

If your child says… “I only need to change the two ends.”

Ask together

If the middle stays unchanged, have you subtracted equally from both comparisons?

What’s true

The middle belongs to both comparisons, so it must receive the same operation.

If your child says… “Passing either condition is enough.”

Ask together

Does three pass the condition ‘twice x is less than six’?

What’s true

Every solution must pass both conditions.