Example Question - number line distance
Here are examples of questions we've helped users solve.
Solving Absolute Value Inequalities
To solve the inequality \( |6 - w| < 59 \), we approach it by considering the definition of the absolute value. The absolute value \( |a| \) of a number is its distance from zero on the number line, regardless of direction. This means \( |6 - w| < 59 \) tells us that the expression \( 6 - w \) must be less than 59 units away from 0 on the number line. We must consider two cases due to the absolute value: 1. \( 6 - w \) is positive or zero: \[ 6 - w < 59 \] \[ -w < 59 - 6 \] \[ -w < 53 \] \[ w > -53 \] (When we divide or multiply by a negative number, we reverse the inequality sign). 2. \( 6 - w \) is negative: \[ -(6 - w) < 59 \] \[ w - 6 < 59 \] \[ w < 59 + 6 \] \[ w < 65 \] Therefore, combining both cases, the solution to the inequality is: \[ -53 < w < 65 \] This tells us w must be greater than -53 and less than 65.
Solving Absolute Value Inequalities
To solve the inequality |u + 6| ≥ 46, we have to consider the definition of absolute value. The expression |u + 6| represents the distance of u + 6 from zero on the number line, and this distance is greater than or equal to 46. We can break this into two separate cases, one for u + 6 being positive and one for u + 6 being negative. Case 1 (u + 6 is positive or 0): u + 6 ≥ 46 u ≥ 46 - 6 u ≥ 40 Case 2 (u + 6 is negative): -(u + 6) ≥ 46 -u - 6 ≥ 46 -u ≥ 46 + 6 -u ≥ 52 Multiply both sides by -1 and reverse the inequality sign (because multiplying by a negative number reverses the inequality): u ≤ -52 So the solution to the inequality |u + 6| ≥ 46 is: u ≥ 40 or u ≤ -52
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com