Problem 50 - Algebra 2

Question

Solve each inequality algebraically. $$ x^{2}+2 x-86 \geq-23 $$

Step-by-Step Solution

Answer

The solution to the inequality is \([-9, 7] \cup [7, \infty)\).

Step 1: Bring the inequality to a standard form

Start by rewriting the inequality so that all terms are on one side of the inequality. We have:\[ x^2 + 2x - 86 \geq -23 \]Add 23 to both sides:\[ x^2 + 2x - 63 \geq 0 \]

Step 2: Factor the quadratic expression

Now factor the quadratic expression \( x^2 + 2x - 63 \).We are looking for two numbers that multiply to -63 and add up to 2. These numbers are 9 and -7.So, we factor the expression as:\[ (x + 9)(x - 7) \geq 0 \]

Step 3: Identify critical values

The critical points from the factors are where each factor equals zero:- \( x + 9 = 0 \Rightarrow x = -9 \)- \( x - 7 = 0 \Rightarrow x = 7 \).These values divide the number line into different intervals.

Step 4: Test intervals

Test each interval determined by the critical values to see where the inequality holds. The intervals are:- \(( -\infty, -9) \)- \([ -9, 7] \)- \((7, \infty ) \)Choose a test point from each interval and substitute back into the inequality:- For \( x = -10 \) in \(( -\infty, -9) \), \((x + 9)(x - 7) < 0 \).- For \( x = 0 \) in \([ -9, 7] \), \((x + 9)(x - 7) > 0 \).- For \( x = 8 \) in \((7, \infty ) \), \((x + 9)(x - 7) > 0 \).Thus, the inequality holds for \([ -9, 7] \) and \((7, \infty ) \).

Step 5: Verify endpoints

Check the endpoints to verify if they satisfy the inequality:- At \(x = -9\), \((x + 9)(x - 7) = 0\).- At \(x = 7\), \((x + 9)(x - 7) = 0\).Since both expressions equal zero, the endpoints are included.Thus, the solution is \([ -9, 7] \, \cup \, [7, \infty )\).

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve quadratic expressions where the terms are related by symbols such as \(\geq\), \(>\), \(\leq\), or \(<\). These inequalities aim to find the range of values for the variable that make the inequality true. For the exercise provided, we started with \(x^{2}+2x-86 \geq -23\). This quadratic inequality involves squaring the variable, but unlike equations, it defines a range of possible solutions rather than just one specific solution. This difference makes solving inequalities a unique challenge. We move all terms to one side of the inequality to bring it into a standard form, resulting in \(x^2 + 2x - 63 \geq 0\). This standard setup helps us proceed with the process of factoring and finding intervals where the inequality holds true.

Factoring Quadratics

Factoring quadratic expressions is a crucial step in solving quadratic inequalities. It's the process of breaking down the quadratic expression into simpler factors that, when multiplied together, give back the original expression. For equations like \(x^2 + 2x - 63\), we look for two numbers that multiply to the constant term, \(-63\), and add up to the middle coefficient, \(2\). In this exercise, those numbers are \(9\) and \(-7\). By factoring, we rewrite the quadratic as \((x + 9)(x - 7)\). This factored form is vital, as it helps us identify critical points, which are the solutions to the equation where each factor is zero. Understanding factoring will enable students to simplify the problem by dealing with each factor separately.

Critical Points

Critical points are values that make each factor of the quadratic expression equal to zero. These points are crucial because they are the boundaries where the inequality may change from true to false. For the factors \((x + 9)(x - 7)\), we set each factor equal to zero to find the critical points:

• \(x + 9 = 0\) leads to \(x = -9\)
• \(x - 7 = 0\) leads to \(x = 7\)

These critical values divide the number line into intervals, demarcating regions where the inequality could behave differently. By analyzing these points and intervals, we can better understand where the inequality holds true. Thus, critical points serve as turning points for the direction of the inequality.

Test Intervals

After determining the critical points, the next step is to test the intervals on the number line they create. Testing intervals helps us determine where the inequality is satisfied. The intervals for our critical points \(-9\) and \(7\) are:

• \((-\infty, -9)\)
• \([-9, 7]\)
• \((7, \infty)\)

We choose test points from each interval, such as \(-10\), \(0\), and \(8\), and substitute them into the inequality \((x + 9)(x - 7) \geq 0\). This tells us whether the inequality holds in that interval. For instance, for \(x = 0\), the result is positive, confirming that the inequality is met in the interval \([-9, 7]\). By systematically checking each segment, students gain a clearer picture of the solution set. This methodical approach ensures accuracy in solving quadratic inequalities.