Solving And Graphing Inequalities Expressing Solutions In Interval Notation

Hey guys! Today, we are diving into the exciting world of inequalities. Inequalities are like equations, but instead of an equals sign, they use symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). We're going to tackle two problems that involve solving inequalities, graphing their solutions on a number line, and expressing those solutions in interval notation. Interval notation is a super handy way to represent a range of numbers, and it's something you'll use a lot in higher-level math. So, buckle up, grab your pencils, and let's get started!

Problem 1: $\frac{3(2 x+16)}{-8} \geq x-1$

Solving the Inequality

Okay, so our first inequality is $\frac{3(2 x+16)}{-8} \geq x-1$. The goal here is to isolate x on one side of the inequality. To do this, we'll follow a series of algebraic steps, just like we would with an equation. However, there's one crucial difference: when we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality sign. Keep that in mind – it’s a common pitfall!

Let's break it down step by step:

1. Simplify the left side: First, let's simplify the left side of the inequality. We have $\frac{3(2 x+16)}{-8}$. Distribute the 3 inside the parentheses:

   $\frac{6x + 48}{-8} \geq x - 1$

2. Divide by -8: Now, divide both terms in the numerator by -8:

   $\frac{6x}{-8} + \frac{48}{-8} \geq x - 1$

   This simplifies to:

   $-\frac{3}{4}x - 6 \geq x - 1$

3. Multiply both sides by -8: To get rid of the fraction, we'll multiply both sides of the inequality by -8. Remember the golden rule: because we're multiplying by a negative number, we need to flip the inequality sign: $(-8) * (-\frac{3}{4}x - 6) \leq (-8) * (x - 1)$

   This gives us:

   $6x + 48 \leq -8x + 8$

4. Get x terms on one side: Next, we want to get all the x terms on one side of the inequality. Let's add 8x to both sides:

   $6x + 8x + 48 \leq -8x + 8x + 8$

   Which simplifies to:

   $14x + 48 \leq 8$

5. Isolate the x term: Now, subtract 48 from both sides to isolate the x term:

   $14x + 48 - 48 \leq 8 - 48$

   This simplifies to:

   $14x \leq -40$

6. Solve for x: Finally, divide both sides by 14 to solve for x:

   $\frac{14x}{14} \leq \frac{-40}{14}$

   This gives us:

   $x \leq -\frac{20}{7}$

So, our solution is $x \leq -\frac{20}{7}$. That means x can be any number less than or equal to -20/7.

Graphing the Solution

Now that we've solved the inequality, let's graph the solution on a number line. This will give us a visual representation of all the values x can take.

1. Draw a number line: Start by drawing a horizontal line. Mark zero in the middle and then add some numbers to the left (negative numbers) and right (positive numbers).

2. Locate -20/7: The number -20/7 is approximately -2.86. Find this point on your number line. It will be a bit before -3.

3. Use a closed circle or bracket: Because our inequality is $x \leq -\frac{20}{7}$ (less than or equal to), we use a closed circle (or a square bracket if you prefer that notation) at -20/7. This indicates that -20/7 is included in the solution.

4. Shade to the left: Since we want all values of x that are less than -20/7, we shade the number line to the left of -20/7. This shaded region represents all the numbers that satisfy the inequality.

Writing the Solution in Interval Notation

Okay, we've solved the inequality and graphed it. Now, let's express the solution in interval notation. Interval notation is a concise way to represent a range of numbers using brackets and parentheses.

1. Left endpoint: The leftmost point in our shaded region is negative infinity (-∞). We always use a parenthesis with infinity because infinity isn't a specific number; it's a concept.

2. Right endpoint: The rightmost point in our shaded region is -20/7. Because -20/7 is included in the solution (we used a closed circle), we use a square bracket.

3. Write the interval: Putting it all together, the solution in interval notation is: $(-\infty, -\frac{20}{7}]$. This means the solution includes all numbers from negative infinity up to and including -20/7.

Problem 2: $-\frac{5}{4}(24-6 m)>14-\frac{1}{2}(16-7 m)$

Solving the Inequality

Alright, let's move on to our second inequality: $-\frac{5}{4}(24-6 m)>14-\frac{1}{2}(16-7 m)$. This one looks a bit more complex, but don't worry, we'll tackle it step by step. Our goal remains the same: isolate m on one side of the inequality.

Here’s the breakdown:

1. Distribute: First, distribute the -5/4 on the left side and the -1/2 on the right side:

   $-\frac{5}{4} * 24 + -\frac{5}{4} * (-6m) > 14 - \frac{1}{2} * 16 - \frac{1}{2} * (-7m)$

   Simplifying this gives us:

   $-30 + \frac{15}{2}m > 14 - 8 + \frac{7}{2}m$

2. Combine like terms: Combine the constants on the right side:

   $-30 + \frac{15}{2}m > 6 + \frac{7}{2}m$

3. Get m terms on one side: Let's subtract (7/2)m from both sides to get the m terms on the left:

   $-30 + \frac{15}{2}m - \frac{7}{2}m > 6 + \frac{7}{2}m - \frac{7}{2}m$

   This simplifies to:

   $-30 + \frac{8}{2}m > 6$

   Which further simplifies to:

   $-30 + 4m > 6$

4. Isolate the m term: Add 30 to both sides to isolate the term with m:

   $-30 + 30 + 4m > 6 + 30$

   This gives us:

   $4m > 36$

5. Solve for m: Finally, divide both sides by 4 to solve for m:

   $\frac{4m}{4} > \frac{36}{4}$

   So, we get:

   $m > 9$

Our solution is $m > 9$. This means m can be any number greater than 9.

Graphing the Solution

Now, let's graph this solution on a number line:

1. Draw a number line: Draw a horizontal line and mark zero in the middle. Add some numbers to the left and right.

2. Locate 9: Find 9 on your number line.

3. Use an open circle or parenthesis: Because our inequality is $m > 9$ (strictly greater than), we use an open circle (or a parenthesis if you prefer) at 9. This indicates that 9 is not included in the solution.

4. Shade to the right: Since we want all values of m that are greater than 9, we shade the number line to the right of 9. This shaded region represents all the numbers that satisfy the inequality.

Writing the Solution in Interval Notation

Let's express our solution in interval notation:

1. Left endpoint: The leftmost point in our shaded region is 9. Because 9 is not included (we used an open circle), we use a parenthesis.

2. Right endpoint: The shaded region extends to positive infinity (+∞). We always use a parenthesis with infinity.

3. Write the interval: The solution in interval notation is: $(9, \infty)$. This means the solution includes all numbers from 9 (but not 9 itself) all the way to positive infinity.

And there you have it! We've successfully solved two inequalities, graphed their solutions on a number line, and expressed those solutions in interval notation. Remember, the key to solving inequalities is to follow the same algebraic steps as with equations, but always flip the inequality sign when multiplying or dividing by a negative number. Graphing the solution helps visualize the range of possible values, and interval notation provides a concise way to represent that range. Keep practicing, and you'll become a pro at solving inequalities in no time! Great job, guys!