Example 1
Completion requirements
| Example 1 |
Multimedia |
Solve the inequality \(x^2 + 10x \ge -24\) using sign analysis. Plot the solution set on a number line.
Start by rearranging the inequality, such that one side is \(0\). Factor the non-zero expression.
\(\begin{align}
x^2 + 10x &\ge - 24 \\
x^2 + 10x + 24 &\ge 0 \\
\left( {x + 4} \right)\left( {x + 6} \right) &\ge 0
\end{align}\)
If you are unsure when a factor is positive, solve the corresponding inequality. For example, \(x + 4\) is positive when \(x + 4 \gt 0\).
\(\begin{align}
x + 4 &\gt 0 \\
x &\gt - 4
\end{align}\)
So the interval \(x \gt -4\) gives positive values for the factor \(x + 4\).
Now, draw a third number line representing the product. Notice that the lines used to separate the positive and negative values in the first two number lines have been extended to show the three potential solution intervals in the product number line.
In the left interval, both factors are negative, so the product is positive. In the middle interval, the first factor is negative and the second is positive, so the product is negative. In the right interval, both factors are positive, so the product is positive.
The expression \((x + 4)(x + 6)\) is positive or zero when \(x \le -6\) or \(x \ge -4\), so the solution set is \(\left\{ {x | x \le - 6 \thinspace \rm{or} \thinspace x \ge -4, \thinspace x \in \rm{R}} \right\}\). Sketch this on a number line.
\(\begin{align}
x^2 + 10x &\ge - 24 \\
x^2 + 10x + 24 &\ge 0 \\
\left( {x + 4} \right)\left( {x + 6} \right) &\ge 0
\end{align}\)
Next, determine the intervals where each factor is positive and negative. Sketching this on a number line is helpful.
If you are unsure when a factor is positive, solve the corresponding inequality. For example, \(x + 4\) is positive when \(x + 4 \gt 0\).
\(\begin{align}
x + 4 &\gt 0 \\
x &\gt - 4
\end{align}\)
So the interval \(x \gt -4\) gives positive values for the factor \(x + 4\).
Now, draw a third number line representing the product. Notice that the lines used to separate the positive and negative values in the first two number lines have been extended to show the three potential solution intervals in the product number line.
In the left interval, both factors are negative, so the product is positive. In the middle interval, the first factor is negative and the second is positive, so the product is negative. In the right interval, both factors are positive, so the product is positive.
The expression \((x + 4)(x + 6)\) is positive or zero when \(x \le -6\) or \(x \ge -4\), so the solution set is \(\left\{ {x | x \le - 6 \thinspace \rm{or} \thinspace x \ge -4, \thinspace x \in \rm{R}} \right\}\). Sketch this on a number line.