US High School Absolute Value Equations and Inequalities — practice questions

1. Mark the number on the number line and count the units back to zero. 2. That count is the absolute value, and it never comes out negative. 3. A number and its opposite sit the same distance away, so they share an absolute value. _Source: OpenStax Intermediate Algebra (CC BY 4.0), Ch 2 "Solving Linear Equations", section 2.7 Solve Absolute Value Inequalities_

1. The two equivalent equations are z equal to 0 and z equal to negative 0. 2. Negative 0 and 0 are the same number. 3. So both branches collapse onto the single value 0. 4. A positive right side would split into two distinct answers, but 0 does not. _Source: OpenStax Intermediate Algebra (CC BY 4.0), Ch 2 "Solving Linear Equations", section 2.7 Solve Absolute Value Inequalities_

1. The absolute value is already alone on the left, so no isolating is needed. 2. The expression inside can be 7 units from zero in either direction. 3. Write the two equivalent equations: 2x minus 3 equals 7, and 2x minus 3 equals negative 7. 4. From the first, 2x equals 10, so x is 5. 5. From the second, 2x equals negative 4, so x is negative 2. 6. Checking x equal to 5 gives the absolute value of 7, and x equal to negative 2 gives the absolute value of negative 7, both equal to 7. 7. Answering 5 or negative 5 flips the sign of the answer rather than of the expression inside. 8. Answering 2 or negative 2 solves only for the second case and mislabels it. 9. Answering 5 or 2 keeps two positive roots and misses the negative branch entirely. _Source: OpenStax Intermediate Algebra (CC BY 4.0), Ch 2 "Solving Linear Equations", section 2.7 Solve Absolute Value Inequalities_