Solve \( 5x - 3 \geq 7x - 9,\ x \in \mathbb{R} \).

Solution

The steps involved in solving a linear inequality are similar to the steps used to solve a linear equation, with one exception: when multiplying or dividing both sides of the inequality by a negative value, the inequality condition must be reversed.

Therefore, the solution is \( \{x\mid x \leq 3 ,\ x \in \mathbb{R}\} \).

The solution can also be stated using interval notation: \( x \in (-\infty, 3],\ x \in \mathbb{R} \).

It can also be represented graphically using a real number line as shown:

Solve \( x^2 - 3x \gt 10,\ x \in \mathbb{R} \).

Algebraic Solution

We begin as we do with a quadratic equation: bring all terms of the inequality to one side, leaving zero on the other side. Then, we factor the quadratic.

Now, a product of two factors is positive in two cases.

Case 1. Both factors are positive.

Case 2. Both factors are negative.

OR\(\qquad\quad\)

Therefore, the solution to this quadratic inequality is \(\left\{ x | x \lt -2 \text{ or } x \gt 5,\, x \in \mathbb{R} \right\} \).

In interval notation, this is \(x \in \left( -\infty, -2\right) \cup \left( 5, \infty\right) \). This can also be illustrated using a number line as shown.

Solve \( x^2 - 3x \gt 10,\ x \in \mathbb{R} \).

Graphical Solution