91Ó°ÊÓ

Linear inequalities are similar to linear equations, but instead of an equal sign, they involve inequality symbols like <, >, \(\text{\textless=}\) or \(\text{\textgreater=}\). The process to solve a linear inequality is much like solving an equation, with a key difference being how we treat multiplication or division by a negative number, which reverses the inequality symbol.

For example, in the exercise \(5x - 3 < 7\), we first add 3 to both sides to isolate the \(x\) term. We're effectively maintaining the balance just like in equations, but we're working with inequalities, considering the entire range of numbers that can satisfy the inequality, rather than a single solution. To finalize, we divide by 5, simplifying it to \(x < 2\), indicating that any number less than 2, but not including 2 itself, will satisfy the inequality.

When solving inequalities for integer solutions, we're looking for whole numbers—positive, negative, or zero—that satisfy the inequality. After solving the inequality \(x < 2\), we identify that all integers less than 2 will be solutions. This includes negative numbers and zero.

Integer solutions can be listed out as in the exercise: \(..., -3, -2, -1, 0, 1\). It's important to remember that integers do not include fractions or decimals. Students are often advised to visualize integers on a number line, which can be a particularly helpful tool for grasping the concept of integer solutions.

Real number solutions include all the possible numbers that could satisfy the inequality, not just integers. Real numbers are all numbers on the number line, including integers, fractions, and irrational numbers like \(\sqrt{2}\) and \(\pi\).

For the inequality \(x < 2\), the real number solutions can be expressed as the interval \(-\infty, 2\), which includes every number up to, but not including, 2. This set of solutions is continuous, meaning that there are an infinite number of possible values for \(x\) within this range, and they are infinitely close to each other—another concept that differentiates real number solutions from integer solutions.