Problem 5 - Algebra 2 and Trigonometry

Question

Write the solution set of each inequality if x is an element of the set of integers. \(x^{2}-3 x+2 \leq 0\)

Step-by-Step Solution

Answer

The solution set in integers is \(\{1, 2\}\).

Step 1: Factor the Quadratic Expression

The inequality you're dealing with is a quadratic in the form of \(x^2 - 3x + 2 \leq 0\). Begin by factoring the quadratic expression. You want to find two numbers that multiply to 2 (the constant term) and add to -3 (the coefficient of the linear term). The quadratic can be factored as \((x-1)(x-2)\).

Step 2: Identify the Critical Values

The factored inequality is \((x - 1)(x - 2) \leq 0\). The critical values, where the expression equals zero, are found by setting each factor equal to zero: \(x - 1 = 0\) and \(x - 2 = 0\). This gives critical values \(x = 1\) and \(x = 2\).

Step 3: Test the Intervals

The critical values divide the number line into intervals: \((-\infty, 1)\), \([1, 2]\), and \((2, \infty)\). Test a value from each interval in the inequality \((x - 1)(x - 2) \leq 0\): - For \(x = 0\) (from \((-\infty, 1)\)), the expression is \((0-1)(0-2) = 2\), which is not \(\leq 0\).- For \(x = 1.5\) (from \([1, 2]\)), the expression is \((1.5-1)(1.5-2) = -0.25\), which is \(\leq 0\). - For \(x = 3\) (from \((2, \infty)\)), the expression is \((3-1)(3-2) = 2\), which is not \(\leq 0\).

Step 4: Solution in Integers

Since we are looking for integer solutions, based on the results from Step 3, the integers \(x\) that satisfy the inequality are those within the interval \([1, 2]\). Therefore, the integer solutions are \(x = 1\) and \(x = 2\).

Key Concepts

Factoring Quadratic Expressions

Factoring a quadratic expression is a crucial skill that simplifies solving quadratic inequalities. The given inequality is in the form \(x^2 - 3x + 2 \leq 0\). To factor a quadratic expression, look for two numbers that multiply to the constant term (here, 2) and add up to the coefficient of the linear term (here, -3).

• Multiply: The two numbers should multiply to 2.
• Add: The two numbers should add to -3.

For the expression \(x^2 - 3x + 2\), the numbers -1 and -2 meet these criteria. Factoring it results in \((x-1)(x-2)\). This step transforms a quadratic equation into a product of linear factors, making it easier to identify critical values and test intervals in subsequent steps.

Critical Values and Testing Intervals

Once the quadratic is factored, the next step is to find critical values by setting each factor to zero.

• Set \(x-1 = 0\) which gives \(x = 1\).
• Set \(x-2 = 0\) which gives \(x = 2\).

These critical values, \(x = 1\) and \(x = 2\), divide the number line into three intervals: \((\-\infty, 1)\), \([1, 2]\), and \((2, \infty)\). Testing a value from each interval in the inequality \((x-1)(x-2) \leq 0\) determines the valid solutions. Insert a number from each interval:

• For \(x = 0\) (from \((\-\infty, 1)\)), the result is 2, not meeting \(\leq 0\).
• For \(x = 1.5\) (from \([1, 2]\)), the result is -0.25, meeting \(\leq 0\).
• For \(x = 3\) (from \((2, \infty)\)), the result is 2, not meeting \(\leq 0\).

Thus, the only interval that satisfies the inequality condition is \([1, 2]\).

Integer Solutions

Integer solutions are sought within the tested intervals. Since the acceptable solutions fall in \([1, 2]\), and we're interested in integer solutions, we consider only integer values within this range. The integers \(x\) that fulfill \((x-1)(x-2) \leq 0\) are those precisely at the endpoints of the interval.

• \(x = 1\)
• \(x = 2\)

These values satisfy the inequality, providing a clear and concise solution set. Understanding critical values, intervals, and integer solutions helps in solving similar quadratic inequalities efficiently.