Inequalities are mathematical expressions where both sides are not equal.

RULES OF INEQUALITIES:

rule 1:

• If, p < q and q < d, then p < d
• If, p > q and q > d, then p > d

rule 2:

• If, p > q, then q < p
• If, p < q, then q > p

rule 3:

If p < q, then p + d < q + d

• If p < q, then p − d < q − d
• If p > q, then p + d > q + d, and
• If p > q, then p − d > q − d

rule 4:

• If p < q, and d is positive, then pd < qd
• If p < q, and d is negative, then pd > qd (inequality swaps)

rule 5:

• If p < q then −p > −q
• If p > q, then −p < −q

rule 6:

• If, p < q, then 1/p > 1/q
• If p > q, then 1/p < 1/q

rule 7:

p2 ≥ 0

rule 8:

If p ≤ q, then √p ≤ √q (for p, q ≥ 0)

SIGN SWAP CHART BELOW:

using the following rules, you can solve inequalities. Isolate the variable:

GRAPHING INEQUALITIES

If  ≤ or ≥, its a closed circle and endpoint is included

If < or >, its open circle and endpoint not included

Use open circle at either ∞ or -∞

Draw a line from the endpoint that extends to the right side if the variable is greater than the number.

Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.

INTERVAL NOTATION:

[] for  ≤ or ≥ on a graph: x ≤ 6= [ -∞,6]

{} for < or > ...

for graphing inequalities with 2 variables: y<x+9 etc

graph the line as y=ax+b, then shade the lesser or greater side, if ≤ or ≥, shade the line too. UPPER side of line=greater vice versa

polynomial inequalities can be a polynomial on one side, inequality sign, and 0 on the other side.

remember:

a<x ≤b=(a,b]

vice versa

If an absolute value is in the inequality, you can remove it by adding  ± to the other side of the equation.