Algebraic inequalities are mathematical statements that compare two expressions using inequality symbols. These symbols include greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Inequalities provide a way to represent relationships between quantities and are an essential tool in solving real-world problems.
To solve algebraic inequalities, we use similar techniques as in solving equations. However, there are a few key differences to keep in mind:
When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol needs to be reversed.
If we need to multiply or divide both sides of an inequality by a variable, we need to consider the sign of the variable. If the variable is positive, the direction of the inequality does not change. However, if the variable is negative, the direction of the inequality needs to be reversed.
Let's solve the inequality: 2x - 5 > 7.
Step 1: Add 5 to both sides of the inequality: 2x - 5 + 5 > 7 + 5.
Step 2: Simplify: 2x > 12.
Step 3: Divide both sides by 2: 2x/2 > 12/2.
Step 4: Simplify: x > 6.
Therefore, the solution to the inequality is x > 6.
Remember, when representing the solution on a number line or graph, we use an open circle for < or > and a closed circle for ≤ or ≥.
Happy exploring and solving algebraic inequalities! Keep up the great work in math!