Problem 44 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ -2 x+1>-11 \text { or } x+1>10 $$

Step-by-Step Solution

Answer

x < 6 or x > 9interval notation: (-∞, 6) ∪ (9, ∞)

Step 1: Solve the first inequality

Solve -2x + 1 > -11Subtract 1 from both sides to isolate the term with the variable:\(-2x > -12\)Next, divide both sides by -2. Remember that dividing by a negative number flips the inequality sign:\(x < 6\)

Step 2: Solve the second inequality

Solve x + 1 > 10Subtract 1 from both sides to isolate x:\(x > 9\)

Step 3: Combine the solutions

The solution to the compound inequality -2x + 1 > -11 or x + 1 > 10is the union of the individual solutions. Therefore, the combined solution is:\(x < 6\) or \(x > 9\)

Step 4: Graph the solution set

Graph the solutions on a number line:Shade to the left of 6 with an open circle at 6, and shade to the right of 9 with an open circle at 9. The graph will show two separate sections on the number line.

Step 5: Write the interval notation

Using interval notation, the solution set is:\((-\thank ,6) \cup (9, \thank )\)

Key Concepts

Inequality Solving

Solving inequalities is similar to solving equations, with a few extra rules. For instance, you still isolate the variable by performing operations like addition, subtraction, multiplication, or division. It's important to remember that when you multiply or divide by a negative number, you must flip the inequality sign.

In our given problem, we had two inequalities: -2x + 1 > -11 and x + 1 > 10.

For the first one, we subtracted 1 from both sides to get:
\( -2x > -12 \).
Then, we divided both sides by -2, flipping the inequality sign, and got: \( x < 6 \).

For the second inequality, we simply subtracted 1 from both sides to isolate the variable, giving us: \( x > 9 \).

So, by solving each inequality separately, you'll isolate the variable in each case and arrive at the solution set.

Interval Notation

Interval notation is a way to represent sets of numbers. It uses brackets and parentheses to show the range of values that satisfy an inequality.

- Parentheses \(( )\) indicate that an endpoint is not included (open circle), while brackets \([ ]\) indicate that an endpoint is included (closed circle).

- In our case, the solution \( x < 6 \) translates to the interval \((-\thank , 6)\), and \( x > 9 \) translates to \((9, \thank )\).

- To combine the solutions, we use the union symbol \( \cup \) which means

Graphing Inequalities

Graphing inequalities involves plotting the solution set on a number line.

- You mark the key points (like 6 and 9) on the number line.
- Use open circles to indicate values that are not included in the solution (since it’s less than or greater than but not equal to).
- Shade the regions to the left of 6 and the right of 9, showing all the values that satisfy the inequality.

This visual representation helps understand the range of possible values for the variable.