Example Question - absolute value expression
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Absolute Value Expression Evaluation
La expresión mostrada en la imagen es: - | -7 | Primero vamos a determinar el valor absoluto de -7. El valor absoluto de un número es la distancia que hay desde ese número hasta el 0 en la recta numérica, sin considerar la dirección. Por lo tanto, el valor absoluto de -7 es 7, ya que está a 7 unidades de distancia del 0. Ahora que tenemos | -7 | = 7, sustituimos este valor en la expresión inicial: - | -7 | = -7 Por lo tanto, el valor de la expresión es -7.
Solving an Inequality Involving Absolute Value
To solve the given inequality, \( 8 - 7|-6s| < -6 \), let's first simplify the expression inside the absolute value sign by multiplying 7 and the absolute value of \(-6s\): \[ 8 - 7 \cdot |-6s| < -6 \] Let \( A = |-6s| \), then we have: \[ 8 - 7A < -6 \] Now, let's solve for \( A \) by isolating it on one side: \[ -7A < -6 - 8 \] \[ -7A < -14 \] Divide both sides by -7, and remember to reverse the inequality sign since we are dividing by a negative number: \[ A > 2 \] But \( A \) was defined as \( |-6s| \), so we substitute back in: \[ |-6s| > 2 \] Now we have to consider the two cases due to the absolute value. This inequality splits into two separate inequalities: 1. When the expression inside the absolute value is positive or zero: \[ -6s > 2 \] \[ s < -\frac{1}{3} \] 2. When the expression inside the absolute value is negative: \[ -6s < -2 \] \[ s > \frac{1}{3} \] Since no values of \( s \) can be simultaneously less than \(-\frac{1}{3}\) and greater than \(\frac{1}{3}\), this inequality has no solution. It means there are no values of \( s \) that can satisfy the original inequality \( 8 - 7|-6s| < -6 \).
Solving an Inequality Involving Absolute Value
To solve the inequality \( 8 - 7|1 - 6s| < -6 \), we could follow these steps: 1. First, let's simplify the inequality by removing the constant term '8' from both sides: \[ 8 - 7|1 - 6s| - 8 < -6 - 8 \] \[ -7|1 - 6s| < -14 \] 2. Now, we divide both sides by '-7'. Remember that dividing by a negative flips the inequality sign: \[ \frac{-7|1 - 6s|}{-7} > \frac{-14}{-7} \] \[ |1 - 6s| > 2 \] 3. The inequality \( |1 - 6s| > 2 \) means that the expression inside the absolute value is either greater than 2 or less than -2. We can express this as two separate inequalities: \[ 1 - 6s > 2 \quad \text{or} \quad 1 - 6s < -2 \] 4. To solve \( 1 - 6s > 2 \), subtract '1' from both sides and then divide by '-6': \[ -6s > 1 \] \[ s < -\frac{1}{6} \] 5. To solve \( 1 - 6s < -2 \), subtract '1' from both sides and then divide by '-6': \[ -6s < -3 \] \[ s > \frac{1}{2} \] The solution to the inequality is therefore \( s < -\frac{1}{6} \) or \( s > \frac{1}{2} \).
Solving Absolute Value Inequalities
To solve the inequality \( |u - 7| > 29 \), we need to split it into two separate inequalities because the absolute value expression can represent a positive or negative result: 1. If the expression inside the absolute value, \( u - 7 \), is positive, then \( u - 7 > 29 \). To solve for \( u \), add 7 to both sides of the inequality: \( u - 7 + 7 > 29 + 7 \) \( u > 36 \) 2. If the expression inside the absolute value, \( u - 7 \), is negative, then \( -(u - 7) > 29 \). This is equivalent to \( 7 - u > 29 \). To solve for \( u \), first subtract 7 from both sides: \( 7 - u - 7 > 29 - 7 \) \( -u > 22 \) Next, multiply both sides by -1 to isolate \( u \), and remember to reverse the inequality sign when multiplying or dividing by a negative number: \( -u(-1) < 22(-1) \) \( u < -22 \) Combine both solutions to express the full solution to the original inequality: \( u > 36 \) or \( u < -22 \) Hence, \( u \) is any number greater than 36 or any number less than -22.
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