Problem 53 - Algebra 2

Question

Use a graphing calculator to solve each inequality. \(-4(6 x-3) \leq 60\)

Step-by-Step Solution

Answer

The solution is \(x \geq -2\).

Step 1: Distribute the Constant

First, let's distribute -4 to both terms inside the parentheses. This means multiplying -4 by 6x and -4 by -3. \[-4(6x - 3) = -4 \cdot 6x + -4 \cdot (-3)\]Perform these multiplications to get:\[-24x + 12\]

Step 2: Write the Simplified Inequality

Now that we've distributed the -4, we have a new inequality to work with:\[-24x + 12 \leq 60\]

Step 3: Subtract 12 from Both Sides

To isolate the term with x, subtract 12 from both sides of the inequality:\[-24x + 12 - 12 \leq 60 - 12\]Simplified, this becomes:\[-24x \leq 48\]

Step 4: Divide by -24 and Flip the Inequality Sign

When you divide by a negative number, you must reverse the inequality sign. Divide both sides by -24:\[x \geq \frac{48}{-24}\]Simplifying this gives:\[x \geq -2\]

Step 5: Use a Graphing Calculator

To verify this solution using a graphing calculator, solve \(-24x \leq 48\) by graphing the equation \(y = -24x + 48\) and finding where it crosses the y-axis at y = 0. The inequality \(x \geq -2\) corresponds to values of x where the graph is below the x-axis.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical concepts, especially when solving inequalities. It helps you see where a function crosses the axes, indicating the range of solutions visually. By graphing the equation form of an inequality, you can quickly identify the values of the variable that satisfy it.

Using a graphing calculator involves a few straightforward steps:

• Enter the function into the calculator. In this case, you want to graph the function derived from the inequality: \(y = -24x + 48\).
• Look for the x-intercept where the graph crosses the x-axis (y=0).
• The portion of the x-axis where the graph is below the x-axis aligns with the solution to the inequality.

Graphing calculators make it simple to check your work. You get a quick visual check for correctness, ensuring the solution \(x \geq -2\) is valid.

Distributive Property

The distributive property is a basic algebraic principle that allows you to multiply a constant by terms inside parentheses. This property ensures that each term within the parentheses is multiplied by the term outside. In the case of the inequality \(-4(6x - 3)\),
you apply the distributive property to distribute \(-4\) to both \(6x\) and \(-3\).

This results in:

• \(-4 \times 6x = -24x\)
• \(-4 \times -3 = +12\)

By correctly applying the distributive property, the original inequality simplifies to \(-24x + 12 \leq 60\).

This step paves the way for further manipulation of the inequality. It's essential for making complex expressions manageable and easier to solve.

Solving Inequalities

Solving inequalities involves finding all values of a variable that make the inequality true. It's similar to solving equations, but with one important distinction. When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

The inequality \(-24x + 12 \leq 60\) is solved by isolating \(x\).

• Subtract 12: This step removes the constant part, changing the inequality to \(-24x \leq 48\).
• Divide by -24: Here’s the crucial part. When dividing by a negative, flip \(\leq\) to \(\geq\). The inequality becomes \(x \geq -2\).

Remember this rule, as it's a common place where mistakes happen. Keeping track of inequality signs is essential for producing accurate solutions.

Graphical Solution

A graphical solution lets you interpret inequalities visually. You can directly see the region of solutions without performing exact calculations. By graphing the function \(y = -24x + 48\), you observe where the graph is below the x-axis, indicating the range of x-values that satisfy the inequality.

Here's how you approach it:

• Graph \(y = -24x + 48\) on the coordinate plane.
• Check where the graph meets the x-axis (y=0).
• The inequality solution \(x \geq -2\) corresponds with the graph below the x-axis.

Using visual aids like graphs strengthens your understanding of algebraic solutions. It ensures that you grasp both the theoretical solution and its practical visual confirmation.