Problem 68 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -5 \leq 5(-x+1)<15 $$

Step-by-Step Solution

Answer

The solution set is \((-2, 2]\). Graph: open at -2, closed at 2.

Step 1: Distribute the 5

First, distribute the 5 in the inequality to remove the parentheses. This means multiplying 5 by each term inside the parentheses:\[-5 \leq 5(-x+1) < 15\]Becomes:\[-5 \leq 5(-x) + 5 < 15\]

Step 2: Simplify the expression

Now simplify the expression by multiplying:\[-5 \leq -5x + 5 < 15\].

Step 3: Isolate middle term by subtracting 5

Subtract 5 from all parts of the inequality to isolate the term with the variable:\[-5 - 5 \leq -5x + 5 - 5 < 15 - 5\]Simplifies to:\[-10 \leq -5x < 10\].

Step 4: Solve for x

Divide every part of the inequality by -5. Remember, dividing or multiplying by a negative number reverses the inequality signs:\[ -10 \div (-5) \geq x \geq 10 \div (-5) \]This simplifies to:\[2 \geq x > -2\], or flip it for clarity:\[-2 < x \leq 2\].

Step 5: Graph the solution set

To graph \(-2 < x \leq 2\), draw a number line and mark the intervals:- Open circle on -2 (not including -2).- Closed circle on 2 (including 2).- Shade the region between -2 and 2.

Step 6: Write in interval notation

The interval notation for the solution set \(-2 < x \leq 2\) is \((-2, 2]\).

Key Concepts

Interval Notation

Interval notation offers a concise way to write the range of values found in a set. It replaces "less than" or "greater than" signs and is often used in solutions of inequalities.
When writing interval notation, remember:

• A square bracket \'[\' or \'\']\' indicates that the endpoint is included in the set, known as closed interval.
• A parenthesis \'(\' or \'\')\' signifies that the endpoint is not included, known as open interval.

For example, the inequality solution \(-2 < x \leq 2\) is expressed in interval notation as \((-2, 2]\). Here, \(-2\) is not included in the interval, so a parenthesis is used, whereas \(+2\) is included, requiring a bracket.
This clear formatting makes understanding and interpreting solutions easier, especially when graphing.

Solution Set

The solution set is the collection of all possible values that satisfy the given inequality. Knowing how to find it is fundamental to solving inequalities.
To determine a solution set:

• Start by simplifying the inequality using basic arithmetic rules.
• Isolate the variable to discover all values it can take.
• Consider any restrictions that reverse the inequality, such as multiplying or dividing by negative numbers.

For the inequality \(-5 \leq 5(-x+1) < 15\), transformation steps took us to \: \(-2 < x \leq 2\).
This results in a set of values for \('x'\) that meet the inequality's conditions. \('x'\) can be anything between \(-2\) and \(+2\), not including \(-2\) but including \(+2\). Understanding the solution set gives a complete picture of the inequality's answer.

Graphing Inequalities

Graphing inequalities visually shows the range of possible solutions on a number line, helping to confirm understanding.
To graph an inequality like \(-2 < x \leq 2\):

• Draw a horizontal line segment on a number line covering values from \(-2\) to \(+2\).
• Mark an open circle over \(-2\) indicating that it is not part of the solution set.
• Place a closed circle over \(+2\) indicating that it is included in the solution set.
• Shade the region between \(-2\) and \(+2\) to show all numbers \(x\) satisfying the inequality.

Graphing provides a quick, visual understanding of the range and scope of the solution set, clarifying options for \(x\). It’s a beneficial step before writing interval notation, reinforcing the clarity of the answer.