Question

Solve the inequality. $$|x+10| \geq 20$$

Short answer

The solution to the inequality \(|x+10| \geq 20\) is \(x \geq 10\) and \(x \leq -30\)

Step-by-step solution

Identify the Inequality

The given inequality is \(|x + 10| \geq 20\).

Remove the Absolute Value

This can be done by splitting the inequality into two: \(x + 10 \geq 20\) and \(- (x + 10) \geq 20\)

Solve First Inequality

To solve \(x + 10 \geq 20\), we simply subtract 10 from both sides to obtain: \(x \geq 10\)

Solve Second Inequality

To solve \(- (x + 10) \geq 20\), we multiply all terms by -1. Remembering that when we multiply an inequality by a negative number, the inequality sign flips, we get: \(x + 10 \leq -20\). Subtracting 10 from both sides gives: \(x \leq -30\)

Combine All Solutions

Finally, we combine the solutions from the two inequalities. The solution for this absolute value inequality is thus \(x \geq 10\) and \(x \leq -30\)

Key concepts

Absolute Value

In algebra, the absolute value is defined as the distance of a number from zero on the number line, regardless of direction. The notation used for the absolute value of a number is two vertical bars surrounding the number or expression, as in the current exercise, \( |x+10| \). This means we are looking at how far the quantity \(x+10\) is from zero.

When faced with absolute value inequalities like \( |x+10| \geq 20 \), we understand that we are seeking all the values of \(x\) such that the expression \(x+10\) is at least 20 units away from zero, either in the positive or negative direction on the number line.

Inequalities

Inequalities represent a relation between two expressions that aren't necessarily equal, but instead have a greater-than or less-than relationship. They are represented by signs such as \( > \), \( < \), \( \geq \), and \( \leq \).

In our inequality \( |x+10| \geq 20 \), the greater-than-or-equal-to sign \( \geq \) signifies that the absolute value of \(x+10\) can be greater than or equal to 20. Solving inequalities typically involves finding the set of all possible values of the variable that make the inequality true.

Algebraic Solutions

Algebraic solutions to inequalities involve using algebraic methods to manipulate the expressions to determine the solution set. In the case of absolute value inequalities, the procedure starts by considering two separate cases - one for the positive and one for the negative scenario of what's inside the absolute value.

For the inequality \( |x+10| \geq 20 \), we evaluate the cases \(x+10 \geq 20\) and \(x+10 \leq -20\), effectively removing the absolute value and revealing the solution set through further algebraic simplification. By isolating \(x\) in each case, you obtain the range or ranges of values that satisfy the original inequality.

Inequality Notation

Inequality notation expresses the solution set of an inequality, supplying the boundaries and indicating whether these boundaries are included in the solution set. This is often shown on a number line or with interval notation.

For the inequalities resulting from \( |x+10| \geq 20 \), the solutions \(x \geq 10\) and \(x \leq -30\) express that \(x\) can be any number greater than or equal to 10, or less than or equal to -30. In interval notation, this is represented by \( (-\infty, -30] \cup [10, \infty) \), signifying that \(x\) belongs to one of these two intervals.