What is an inequality in algebra?

An inequality is a mathematical statement that compares two expressions using signs such as <, >, ≤, or ≥. For example, the inequality x + 3 > 5 means that x must be greater than 2.

What is the importance of the inequality signs?

Inequality signs indicate the relationship between two values. For example, if a < b, it means a is less than b. Understanding these signs is crucial for solving inequalities and graphing their solutions.

What is the solution of the inequality 2x - 4 < 6? Hint: Add 4 to both sides and then divide by 2.

First, add 4 to both sides: 2x - 4 + 4 < 6 + 4, simplifying to 2x < 10. Then divide by 2: x < 5. The solution is all x values less than 5.

What is the formula for solving a quadratic inequality?

To solve a quadratic inequality, first solve the corresponding quadratic equation. For example, to solve x² - 5x + 6 < 0, find the roots x = 2 and x = 3. Then test intervals to determine where the inequality holds.

If x² - 4 ≤ 0, what are the values of x? Hint: Factor the expression.

Factor the expression: (x - 2)(x + 2) ≤ 0. The roots are x = -2 and x = 2. Testing intervals gives the solution: -2 ≤ x ≤ 2.

What is a common mistake when solving inequalities?

A common mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. For example, if you have -2x < 4 and divide by -2, the inequality becomes x > -2.

What are the steps to graph the solution of an inequality?

To graph the solution of an inequality, first find the boundary points by solving the equality. Then determine which side of the boundary satisfies the inequality by testing a point. Finally, use open circles for < or > and closed circles for ≤ or ≥.

Solve the inequality 3(x - 1) > 2(x + 2). Hint: Distribute first, then isolate x.

Distributing gives 3x - 3 > 2x + 4. Subtract 2x from both sides: x - 3 > 4. Then add 3: x > 7.

If 4x + 7 ≥ 3x + 10, what is the value of x? Hint: Isolate x by subtracting 3x from both sides.

Subtracting gives 4x - 3x + 7 ≥ 10. Simplifying, x + 7 ≥ 10. Then, subtract 7: x ≥ 3.

What is an inequality in algebra?

An inequality is a mathematical statement that indicates one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. For example, x > 5 means that x can take any value greater than 5.

What is the key property of inequalities when multiplying or dividing by a negative number?

When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if -2x < 4, dividing by -2 gives x > -2.

What is the formula for solving linear inequalities?

To solve a linear inequality in the form ax + b < c, isolate the variable x by performing inverse operations. For example, to solve 3x + 2 < 11, subtract 2 from both sides to get 3x < 9, then divide by 3 to find x < 3.

If 2x - 5 > 3, what is the solution set for x? Hint: Start by adding 5 to both sides.

Add 5 to both sides: 2x - 5 + 5 > 3 + 5, which simplifies to 2x > 8. Then divide by 2: x > 4. The solution set is x > 4.

What are the critical points in the inequality x² - 4 < 0? Hint: Factor the quadratic expression.

Factoring gives (x - 2)(x + 2) < 0. The critical points are x = -2 and x = 2. To find the intervals where the inequality holds, test the intervals formed by these points.

What is the relationship between the number of solutions and the direction of the inequality sign?

For inequalities, the direction of the inequality sign affects the solution set. For example, x < 3 has infinitely many solutions (all values less than 3), while if the inequality was x ≤ 3, the solution set also includes 3.