Question

Solve the inequality: \(x^{2}-4 \leq 5\)

Short answer

−3 ≤ x ≤ 3

Step-by-step solution

Rewrite the Inequality

Start by moving all terms to one side of the inequality to set it to zero. The inequality is originally given as: equation\(x^{2} - 4 \leq 5\). Move 5 to the left side to get: \(x^{2} - 4 - 5 \leq 0\).This simplifies to:\(x^{2} - 9 \leq 0\).

Factor the Quadratic Expression

Next, factor the quadratic expression on the left side of the inequality. This gives:\(x^{2} - 9 = (x - 3)(x + 3)\).So the inequality now is:\((x - 3)(x + 3) \leq 0\).

Determine the Critical Points

Identify the values of x that make the expression zero. Solve \((x - 3)(x + 3) = 0\) to get the critical points:x = 3 and x = -3.

Test Intervals

Determine the sign of the expression in each interval defined by the critical points. The intervals to test are: (-∞, -3), [-3, 3], and (3, ∞). 1. Test a point in (-∞, -3), e.g., x = -4:\((x - 3)(x + 3) = (-4 - 3)(-4 + 3) = (−7)(−1) = 7\) which is positive. 2. Test a point in [-3, 3], e.g., x = 0: \((x - 3)(x + 3) = (0 - 3)(0 + 3) = (−3)(3) = −9\),which is negative. 3. Test a point in (3, ∞), e.g., x = 4:\((x - 3)(x + 3) = (4 - 3)(4 + 3) = (1)(7) = 7\), which is positive. Since we want the expression to be less than or equal to zero, we select the interval [-3, 3].

Write the Solution

The solution to the inequality is the interval where the expression is less than or equal to zero, which we found to be: [-3, 3].Therefore,\(−3 ≤ x ≤ 3\).

Key concepts

Factoring Quadratics

Factoring quadratics is a method used to break down a quadratic expression into simpler components called factors. For the expression \(x^2 - 9\), we look for two numbers that multiply to equal the constant term (-9) and add up to the coefficient of the middle term (0 in this case because there is no middle term). Since \(x^2 - 9\) fits the pattern of a difference of squares (\(a^2 - b^2 = (a - b)(a + b)\)), we can factor it as\((x - 3)(x + 3)\). This simplifies future steps by turning a single quadratic expression into a product of two binomial expressions. These factors are crucial for determining the critical points and testing intervals.

Critical Points

Critical points are the values of x that make the factored quadratic expression equal to zero. To find these points, we solve for x in the equations \((x - 3) = 0\) and \((x + 3) = 0\). Solving these equations:

• From \((x - 3) = 0\): x = 3
• From \((x + 3) = 0\): x = -3

These values split the number line into distinct intervals, which are important for the testing intervals step. Critical points help us understand where the quadratic expression changes its sign (positive to negative or vice versa).

Testing Intervals

Testing intervals involves choosing sample points from the regions divided by the critical points and evaluating the expression at those points. For our inequality, the critical points split the x-axis into three intervals:

• \((-∞, -3)\)
• \([-3, 3]\)
• \((3, ∞)\)

We select a point in each interval to determine whether the expression is positive or negative:

• In \((-∞, -3)\), we tested with x = -4: \((-4 - 3)(-4 + 3) = 7\) (positive)
• In \([-3, 3]\), we tested with x = 0: \((0 - 3)(0 + 3) = -9\) (negative)
• In \((3, ∞)\), we tested with x = 4: \((4 - 3)(4 + 3) = 7\) (positive)

These results show the sign of the expression in each interval, helping us identify where the inequality holds true.

Inequality Solution Steps

The final step is to bring all the results together to solve the inequality. This can be broken down into clear sub-steps per our previous explanations:

• Rewrite the inequality so that one side equals zero: \(x^2 - 9 ≤ 0\)
• Factor the quadratic expression: \((x - 3)(x + 3) ≤ 0\)
• Find the critical points by solving \((x - 3) = 0\) and \((x + 3) = 0\), resulting in x = 3 and x = -3
• Test the intervals \((-∞, -3)\), \([-3, 3]\), and \((3, ∞)\) with points from each interval
• Identify where the expression is less than or equal to zero: \([-3, 3]\)

The solution to the inequality \(x^2 - 9 ≤ 0\) is therefore \(-3 ≤ x ≤ 3\). By following these steps, you ensure that you cover every aspect of solving the inequality systematically. This method not only finds the solution but also helps understand why the solution is correct.