Problem 44 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$1-\frac{5 x}{4} \geq 2 x+13$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq -\frac{48}{13}\). This is represented on the number line as a filled-in circle at -\frac{48}{13}, with an arrow pointing to the left to include all numbers less than or equal to -\frac{48}{13}

Step 1: Combine Like Terms

The inequality is given as \(1 - \frac{5x}{4} \geq 2x + 13\). To start the problem, we need to combine like terms i.e., we take the term containing x on one side and constants on the other side. It will look something like this: \(1 - 13 \geq 2x + \frac{5x}{4}\)

Step 2: Solve for x

Continue by performing the arithmetic operation on the left side that results in \(-12 \geq 2x + \frac{5x}{4}\). Simplify the right side by choosing a common denominator for the x terms and simplifying. This gives \(-12 \geq \frac{8x + 5x}{4}\), which further simplifies to \(-12 \geq \frac{13x}{4}\). To solve for x, we multiply each side by \(\frac{4}{13}\) to give \(-\frac{48}{13} \geq x\) or \(x \leq -\frac{48}{13}\)

Step 3: Graph the Solution

On a number line, we represent the solution \(x \leq -\frac{48}{13}\) as a filled-in circle at -\frac{48}{13}, and an arrow pointing to the left to represent all numbers less than or equal to -\frac{48}{13}

Key Concepts

Solving Inequalities

Inequalities are expressions that describe the relative size or order of two values. They use symbols like \( >, <, \geq, \) and \( \leq \). When solving inequalities, the main goal is to isolate the variable, often "x," on one side. This process is similar to solving equations, but with one important distinction: If you multiply or divide both sides of an inequality by a negative number, the inequality sign flips.
To solve the inequality \(1 - \frac{5x}{4} \geq 2x + 13\), we begin by rearranging so all "x" terms are on one side and constants on the other.

• First, subtract 1 from both sides: \(- \frac{5x}{4} \geq 2x + 12\)
• Then, bring "x" terms to one side by subtracting \(2x\) from both sides: \(-\frac{5x}{4} - 2x \geq 12\)

Next, find a common denominator (which is 4 here) to combine the "x" terms:

• \(-\frac{5x}{4} - \frac{8x}{4} \geq 12\)
• Simplify to \(-\frac{13x}{4} \geq 12\)

Multiply through by \(-4/13\) to solve for "x," remembering to flip the inequality: \(x \leq -\frac{48}{13}\).
It's crucial to understand these steps as they lead you to the correct solution efficiently.

Graphing Inequalities

Graphing inequalities helps visualize the solution set on a number line. For our solution, \(x \leq -\frac{48}{13}\), we use specific symbols and methods to depict the range of "x" values.
Here's how to graph it:

• Draw a horizontal line representing the number line.
• Locate the point \(-\frac{48}{13}\) on this line. Since calculating \(-\frac{48}{13}\) gives approximately \(-3.69\), this point will be slightly to the left of \(-3.5\).
• Place a filled-in circle at \(-\frac{48}{13}\). This indicates the value is included in the solution (because of the "less than or equal to" sign).
• Draw an arrow pointing left from the filled circle. This arrow shows all values less than \(-\frac{48}{13}\) are part of the solution.

Graphing shows not just the solution but also the range of possible values that solve the inequality, making it clearer and more intuitive to understand.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operation symbols. Expressions can vary from simple to complex, comprising terms joined by addition or subtraction.
In our exercise, \(1 - \frac{5x}{4} \geq 2x + 13\), several key components were involved:

• "1" and "13" are constant terms, which are numbers on their own.
• \(-\frac{5x}{4}\) and \(2x\) are terms that incorporate the variable "x," indicating not just quantities but also relationships based on the value of "x."

Understanding how terms work in expression is critical for simplifying and solving them effectively.
When solving equations or inequalities, our main task is to manipulate these expressions to isolate the variable. This often involves combining like terms—terms with the same variable and corresponding power, such as \(-\frac{5x}{4}\) and \(2x\). Merging these intelligently leads to simpler expressions that make finding solutions more straightforward.