Question

Find the range of the values of \(x\) which satisfy the inequation, \(x^{2}-7 x+3<2 x+25\). (1) \((-2,11)\) (2) \((2,11)\) (3) \((-\infty,-1) \cup(2,11)\) (4) \((-8,-2) \cup[11, \infty)\)

Short answer

The values of x that satisfy the inequality are in the open interval \(-2 < x < 11\).

Step-by-step solution

Bring all terms to one side of the inequality

Subtract \(2x + 25\) from both sides of the inequality: \(x^2 - 7x + 3 < 2x + 25 \implies x^2 - 9x - 22 < 0\).

Factor the quadratic inequality

We need to factor the quadratic \(x^2 - 9x -22\): \((x - 11)(x + 2) < 0\).

Identify critical points

The critical points are the roots of the factored expression, which are the values of x that make the expression equal to 0: \(x - 11 = 0 \implies x = 11\) \(x + 2 = 0 \implies x = -2\)

Find the intervals that satisfy the inequality

Check intervals using the critical points we found. We mainly need to check the signs of the factors: 1. \(x < -2: (x - 11)(x + 2) > 0\) 2. \(-2 < x < 11: (x - 11)(x + 2) < 0\) 3. \(x > 11: (x - 11)(x + 2) > 0\) The inequality is true for the interval \(-2 < x < 11\), so the correct answer is: \((2) \; (2, 11)\)

Key concepts

Factoring Quadratics

Understanding how to factor quadratics is a foundation stone of algebra and crucial for solving quadratic inequalities. Factoring involves expressing a quadratic equation, typically in the form of \(ax^2 + bx + c\), as the product of two binomial expressions. Imagine you are working with a jigsaw puzzle. Each piece must fit perfectly to complete the picture, just as each factor in a quadratic must multiply together to become the original quadratic expression.

When factoring the quadratic equation \(x^2 - 9x - 22\), one way to approach it is to find two numbers that multiply to \(-22\) and add up to \(-9\). In our puzzle comparison, these two numbers are like the missing corners of our puzzle that, once found, give us a clear picture. Through trial and error or by applying the AC method (multiplying leading coefficient and the constant term, then finding factors of this product that add up to 'b'), these numbers turn out to be \(-11\) and \(+2\), leading us to the factors \((x - 11)\) and \((x + 2)\).

• Why factoring is important:
• It simplifies the equation for easier understanding.
• It uncovers the roots of the equation, which are the potential critical points.
• It is used to divide the number line into intervals for testing the inequality.

Solving Inequalities

Solving inequalities, like the one in the exercise \(x^2 - 9x - 22 < 0\), is similar to solving equations; however, instead of finding an exact value, the goal is often to determine a range of values that satisfies the inequality. Consider inequalities as the search for a habitat range, where an animal can safely live, just as we seek the 'safe range' of x values where the inequality holds true.

In our case, after factoring the quadratic inequality, we divide the number line into distinct intervals using the critical points \(-2\) and \(11\). We then perform a sign analysis to determine if the factored expression, \((x - 11)(x + 2)\), is less than zero within those intervals. It's like testing the temperature in different areas to see where it's comfortably warm.

• Steps for solving quadratic inequalities:
  • Rearrange the inequality so that all terms are on one side and zero is on the other.
  • Factor the quadratic equation if possible.
  • Identify and plot the critical points on a number line.
  • Test the intervals between critical points to determine where the inequality is satisfied.

Through this process, we determine that \(x\) must be between \(-2\) and \(11\) for the inequality to be true, but not including \(-2\) and \(11\) since the inequality is strict (<), not inclusive (\(\leq\) or \(\geq\)).

Critical Points

Critical points in the context of solving inequalities are values where the expression changes sign. You can think of them as 'border stations' between countries - areas where you move from one 'state' to another. For the quadratic inequality \((x - 11)(x + 2) < 0\), the critical points are the solutions to \(x - 11 = 0\) and \(x + 2 = 0\), which are \(x = 11\) and \(x = -2\), respectively.

These points divide the number line into different regions where the quadratic expression will take on different signs. Critical points are like checkpoints; we examine what happens as we 'cross' them:

• If we choose a test value less than \(-2\), both factors \((x - 11)\) and \((x + 2)\) are negative, which makes the product positive, not meeting our inequality condition.
• Between \(-2\) and \(11\), the factor \((x - 11)\) is negative and \((x + 2)\) is positive, giving a negative product, which is what we want.
• If we choose a test value greater than \(11\), both factors are positive, giving a product that's positive and not satisfying the inequality.

Identifying and testing around critical points is essential, as it tells us precisely where the inequality changes its nature. For real-life situations, understanding such points can help in optimizing or avoiding certain ranges, much like car navigation systems that find the best route by recognizing where traffic patterns change.