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Absolute Value Inequalities. We begin by examining the solutions to the following inequality: \(| x | \leq 3\) The absolute value of a number represents the distance from the origin. Therefore, this equation describes all numbers whose distance from zero is less than or equal to \(3\). We can graph this solution set by shading all such numbers.
The property of absolute value tells us that |a| = a ∣a∣ = a for non-negative a a, so in this case |x+3|<7 \implies x+3 < 7\implies x<4. ∣x+3∣ < 7 x+ 3 < 7 x < 4. Now, we have two inequalities here, and the solution for this case is the intersection of both inequalities.
4 de jun. de 2023 · property 1. The solution of |x| < a depends upon the value and sign of a. Case I: a < 0. The inequality |x| < a has no solution. Case II: a = 0. The inequality |x| < 0 has no solution. Case III: a > 0. The inequality |x| < a has solution set {x : −a < x < a}. Let’s look at some examples. Example 4.4.1 4.4. 1. Solve the inequality |x| < −5 for x.
To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently. Definition: Absolute Value Inequalities. For an algebraic expression X and k > 0, an absolute value inequality is an inequality of the form.
To solve inequalities with absolute values, use a number line to see how far the absolute value is from zero. Split into two cases: when it is positive or negative. Solve each case with algebra. The answer is both cases together, in intervals or words. Created by Sal Khan and CK-12 Foundation.
Absolute Value Inequalities. For an algebraic expression X, and k > 0, an absolute value inequality is an inequality of the form. |X| < k is equivalent to − k < X < k |X| > k is equivalent to X < − k or X > k. These statements also apply to |X| ≤ k and |X| ≥ k.
This topic covers: Solving absolute value equations. Graphing absolute value functions. Solving absolute value inequalities.
