Equations and inequalities

Understanding Equations

An equation is a mathematical statement that asserts the equality of two expressions. It includes an equals sign (=).
The main goal in solving an equation is to determine what values of the variable make the equation true.
For example, in the equation 2x + 3 = 9, we could solve for x by subtracting 3 from both sides to get 2x = 6, and then dividing both sides by 2 to find that x = 3.

Solving Linear Equations

Linear equations are equations in which none of the variables are raised to a power higher than 1.
Solving linear equations generally involves isolating the variable on one side of the equals sign.
For instance, the equation 5x-7=18 would become 5x=25 by adding 7 to both sides, and finally x = 5 by dividing both sides by 5.

Solving Quadratic Equations

A quadratic equation is an equation that can be rewritten in the format ax^2 + bx + c = 0, where a, b, and c are constants.
Quadratic equations can be solved by factoring, completing the square, using the quadratic formula, or graphing.
The quadratic formula is given by x = [ -b ± sqrt(b^2 - 4ac)] / 2a and can be used when other methods are too complex.

Understanding Inequalities

An inequality is a statement that shows a relationship between two mathematical expressions that may not be equal.
Inequalities use symbols like <, >, ≤, and ≥ to show the relationship between the expressions.
For instance, the inequality x + 2 > 3 means that for the statement to be true, x must be greater than 1.

Solving Linear Inequalities

Solving linear inequalities is similar to solving linear equations, but instead of finding a specific value for the variable, you’ll find a range of values.
Be aware that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.
For example, in the inequality -2x ≤ 4, if we divide both sides by -2, remembering to flip the inequality sign, we find x ≥ -2. This means all values of x that are greater than or equal to -2 will satisfy this inequality.

Solving Quadratic Inequalities

Solving quadratic inequalities can be accomplished through methods such as factoring and graphing.
When graphing a quadratic inequality, solutions are the x-values beneath or above the parabola, depending on whether it’s a “greater than” or “less than” inequality.
Remember to take into account whether the inequality is strict (> or <) or includes equality (≥ or ≤) when determining the solution interval.