Problem 10 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ -4 x \geq 10 $$

Step-by-Step Solution

Answer

The solution is \(x \leq -\frac{5}{2}\), or \((-\infty, -\frac{5}{2}]\) in interval notation.

Step 1: Isolate the variable

To isolate the variable \( x \), we begin by dividing all terms in the inequality \(-4x \geq 10\) by \(-4\). Remember, when you multiply or divide both sides of an inequality by a negative number, the inequality symbol flips. Hence, \(x \leq -\frac{10}{4}\).

Step 2: Simplify the expression

Simplify \(-\frac{10}{4}\) to its simplest form by dividing the numerator and the denominator by their greatest common divisor, which is 2. This gives us \(x \leq -\frac{5}{2}\).

Step 3: Express in interval notation

The solution \(x \leq -\frac{5}{2}\) is expressed in interval notation as \((-\infty, -\frac{5}{2}]\). It includes all numbers less than or equal to \(-\frac{5}{2}\).

Step 4: Graph the solution set

To graph the solution set, draw a number line and mark a point at \(-\frac{5}{2}\). Since \(x \) includes \(-\frac{5}{2}\) itself, mark this point with a closed dot. Shade the line to the left of \(-\frac{5}{2}\) to represent all numbers less than or equal to this value.

Key Concepts

Interval Notation

Interval notation is a way to express a set of numbers along a continuum, often derived from inequalities. It is especially useful when dealing with linear inequalities, such as those involving greater than or less than signs. In our case, we arrived at the inequality \(x \leq -\frac{5}{2}\).
Interval notation translates this into \(( -\infty, -\frac{5}{2}]\). This means that the set of solutions includes all numbers up to and including \(-\frac{5}{2}\), but not beyond.

This notation uses the following symbols:

• Parentheses \

Inequality Symbol

An inequality symbol is a mathematical notation that denotes the relative size or order of two values. In inequalities, you typically encounter four primary symbols: ">", "<", "≥", and "≤".
When solving linear inequalities like \(-4x \geq 10\), the inequality involves the "≥" symbol, meaning "greater than or equal to". This indicates that one side of the inequality is not only greater than the other side but could also be equal to it.

Important note: When an inequality involves multiplication or division by a negative number, the symbol's direction must be flipped. This was crucial in our exercise, resulting in \(x \leq -\frac{5}{2}\). This reflects the switch to a "less than or equal to" relationship due to division by -4.
Understanding these symbols helps in correctly interpreting the solution and ensuring the inequality's conditions are properly met.

Solving Inequalities

Solving inequalities involves finding the range of values that make an inequality true. The process is similar to solving equations but with an important exception related to inequality symbols.
Here's how to solve the inequality \(-4x \geq 10\):

• Isolate the Variable: Begin by getting the variable on one side. Divide both sides by -4 to achieve \(x \leq -\frac{10}{4}\). Note the symbol changes because of the division by a negative number.
• Simplify: Simplify to reach \(x \leq -\frac{5}{2}\). This involves dividing the fraction further to reach its simplest form.

By understanding these steps, you'll be able to tackle similar inequalities with confidence, ensuring you don't overlook key rules, such as the flipping of the inequality symbol.

Graphing Inequalities

Graphing inequalities is a visual way to represent the solutions of an inequality on a number line or coordinate plane. This helps in seeing the range of potential solutions in a clear, graphical form.
To graph the solution set of \(x \leq -\frac{5}{2}\):

• Draw the Number Line: Begin with a horizontal line and choose appropriate scale markings.
• Mark the Point: Locate \(-\frac{5}{2}\) on the number line. Use a closed dot since it is included (corresponding to ≤ symbol).
• Shade the Solution Area: Shade to the left of \(-\frac{5}{2}\) to indicate all numbers less than or equal to \(-\frac{5}{2}\).

Graphing is beneficial because it provides a quick, visual method to verify the solution of the inequality. It shows not just the specific solution but the entire range of solutions visually.