Problem 45 - Understanding Elementary Algebra with Geometry

Question

Solve the given inequality and sketch the solution set on a number line. $$9-5(x+4)>9$$

Step-by-Step Solution

Answer

\(x < -4\)

Step 1: Distribute the term

To start solving the inequality, distribute the -5 through the parentheses.\[9 - 5(x + 4) > 9\]Distributing gives:\[9 - 5x - 20 > 9\]

Step 2: Simplify the inequality

Combine like terms on the left side.\[9 - 20 - 5x > 9\]This simplifies to:\[-11 - 5x > 9\]

Step 3: Isolate the variable term

Add 11 to both sides to isolate the term with the variable.\[-11 - 5x + 11 > 9 + 11\]This becomes:\[-5x > 20\]

Step 4: Solve for the variable

To solve for \(x\), divide both sides by -5. Remember to reverse the inequality sign when dividing by a negative number.\[x < \frac{20}{-5}\]So,\[x < -4\]

Step 5: Sketch the solution set on a number line

The solution \(x < -4\) means that all numbers less than -4 satisfy the inequality. This is shown on a number line with an open circle at -4 and shading to the left.--|---|---|---●====>-5 -4

Key Concepts

Distributing Terms

Distributing terms, also known as the distributive property, helps break down expressions within parentheses to make the equation easier to manage. In the context of inequalities like the given one, \(9 - 5(x + 4) > 9\), you need to distribute \(-5\) across \(x + 4\).

After applying the distributive property, you get:

• \( -5 \times x = -5x \)
• \( -5 \times 4 = -20 \)

So, \( 9 - 5(x + 4) \) becomes \( 9 - 5x - 20 \). This way, the equation is simplified to \( 9 - 5x - 20 > 9 \). Distributing terms sets the stage for simplifying your inequality.

Combining Like Terms

Simplifying an inequality often involves combining like terms. This means you combine all the constants (numbers without variables) and all the terms with the same variables. For the given inequality: \( 9 - 5x - 20 > 9 \), you need to combine the constants on the left side:

• The constants are \( 9 \) and \(-20\).

Combining them gives:

\( 9 - 20 = -11 \).

This reduces our inequality to \( -11 - 5x > 9 \). Combining like terms reduces complexity and brings you closer to isolating the variable.

Isolating Variables

Isolating variables is a key step in solving inequalities. It involves moving all terms with the variable to one side of the inequality, and constants to the other side. For the simplified inequality \( -11 - 5x > 9 \), we need to get \(-5x\) by itself on one side:

• Add \( 11 \) to both sides of the inequality.

This results in:

\( -11 + 11 - 5x > 9 + 11 \), which simplifies to \( -5x > 20 \). Isolating the variable prepares the equation for solving \( x \).

Inequality Signs

Inequality signs indicate the relationship between two expressions. Important symbols are:

• '>' means 'greater than'.
• '<' means 'less than'.
• '≥' means 'greater than or equal to'.
• '≤' means 'less than or equal to'.

When solving inequalities, be careful with these signs. For instance, dividing both sides of \(-5x > 20\) by \(-5\) flips the inequality sign. Hence:

\( x < -4 \).

Flipping the inequality sign is crucial when multiplying or dividing both sides by a negative number.

Number Line Representation

To visualize solutions, it's helpful to sketch them on a number line. The solution \( x < -4 \) means any number less than -4 satisfies the inequality. On a number line:

• Place an open circle at -4 (indicating -4 is not included).
• Shade to the left (showing all numbers less than -4).

This graphical representation helps you see the range of values \( x \) can take. Always use an open circle for '<' or '>', and a closed circle for '≤' or '≥'.