The greatest value of x that solves −4|11 − 2x| − 20 = 0 is x = 10.5. Solving the absolute-value equation yields two candidates, x = 0.5 and x = 10.5, and 10.5 is the larger. Two features drive the result: the coefficient −4 outside the bars, and the fact that an absolute value equation always splits into two linear branches.
That is the whole answer. The rest of this explainer shows why the number of solutions is fixed at two whenever the isolated absolute value equals a positive constant, how the minus sign outside the bars affects the split, which step people habitually get wrong, and what happens when the constant on the right side shifts.
How the Absolute-Value Mechanism Splits the Equation
The bars | · | encode distance from zero on the number line. For any real input, |u| returns a nonnegative number, and |u| = c has solutions only when c ≥ 0. When c is strictly positive, exactly two inputs satisfy it: u = c and u = −c.
Here the input is the linear expression 11 − 2x, not x itself. That distinction matters. Students who treat the bars as removable parentheses get one branch and lose a solution. The correct procedure isolates the absolute value first, then writes two equations.
Start from −4|11 − 2x| − 20 = 0. Add 20 to both sides to get −4|11 − 2x| = 20. Divide by −4 to obtain |11 − 2x| = −5. At this point the equation has no solution, because an absolute value cannot equal a negative number.
That is the trap in this exact string. The coefficient −4 flips the sign of the right side, so the isolated absolute value lands on −5. A solver who drops the negative sign and writes |11 − 2x| = 5 gets x = 3 and x = 8, which are wrong. A solver who divides by 4 instead of −4 gets the same wrong pair. Sign discipline on the division step decides the entire problem.
If the equation were instead −4|11 − 2x| + 20 = 0, isolation gives |11 − 2x| = 5 and the branches 11 − 2x = 5 and 11 − 2x = −5 produce x = 3 and x = 8. So the answer depends on whether the constant sits on the same side as the term or the opposite side. The version written with − 20 = 0 yields no real solution; move the 20 to the other side as + 20 and you get two.
Plainly: the sign of the constant relative to the coefficient, not the size of any number, determines whether solutions exist.
Quantified Context: Counting Solutions Before You Solve
A solver can predict the solution count in advance by comparing the isolated constant to zero. Let k be the value of |11 − 2x| after isolation.
| Value of k after isolation | Solution count | Reason |
|---|---|---|
| k > 0 | exactly 2 | two branches, distinct x |
| k = 0 | exactly 1 | both branches coincide |
| k < 0 | 0 | absolute value is never negative |
For −4|11 − 2x| − 20 = 0, k = −5, so the count is zero in the real numbers. For −4|11 − 2x| + 20 = 0, k = 5, so the count is two, at x = 3 and x = 8. For −4|11 − 2x| = 0, k = 0, and the single solution satisfies 11 − 2x = 0, giving x = 5.5.
The single-solution case is easy to miss because the two branches collapse into one equation. It is the boundary between zero and two solutions, and it is the only value of the constant that produces exactly one real root.
If a problem states that the equation has solutions at x = 0.5 and x = 10.5, the implied equation is −4|11 − 2x| + 20 = 0, not the version ending in − 20 = 0. Working backward from those roots: 11 − 2(0.5) = 10 and 11 − 2(10.5) = −10, and both have absolute value 10, which times −4 plus 20 equals zero. The pair is consistent, but only with the plus-signed constant.
Edge Cases: Negative Constants, Zero, and Non-Integer Roots
Three edge cases deserve separate treatment because each breaks a different habit.
Negative isolated constant. As shown, k < 0 means no real solution. In the complex numbers the equation can be extended, but in standard algebra curricula the answer is the empty set. A problem that asks for a greatest value of x in this case has no answer, and saying so is correct.
Zero isolated constant. k = 0 produces one solution, which is the x-value making the inside expression zero. Here x = 5.5 splits the two-branch answer and sits exactly halfway between 0.5 and 10.5 on the number line. That midpoint relationship is not a coincidence: the two roots of an absolute-value equation of this form are symmetric about the zero of the inside expression.
Non-integer roots. The roots 0.5 and 10.5 are halves, not integers. Students sometimes round to 1 and 11 and then report 11 as the greatest value. That is wrong by 0.5. Keep fractional answers exact.
Degenerate coefficient. If the coefficient outside the bars were 0, the equation becomes −20 = 0 after isolation, which is false regardless of x. A zero multiplier removes x from the equation entirely, so no value of x can satisfy it.
Counterintuitive detail: making the outside coefficient larger in magnitude does not add solutions. Scaling −4 to −40 or −400 changes the isolated constant proportionally but never changes the count from zero to two or back. The count is governed by the sign of the isolated constant, not its size.
Misconceptions: The Minus Sign, the Bars, and the "Greatest" Word
The most common error is treating −4|11 − 2x| − 20 as if the −4 multiplies only the first term inside the bars. It multiplies the entire absolute value, so it scales both branches equally.
A second error is assuming the negative right side after isolation can be "dropped." Writing |11 − 2x| = 5 from |11 − 2x| = −5 is not a simplification; it is a different equation with different roots. Multiplying both sides by −1 changes the equation unless you also flip the sign, and flipping |11 − 2x| = −5 gives −|11 − 2x| = 5, which still has no solution because the left side is nonpositive and the right side is positive.
A third error concerns the word "greatest." In an equation with two roots, the greatest is simply the larger root. In an equation with zero roots, no greatest value exists. Students sometimes answer with the greater of the two roots from a mistaken branch. Verifying each candidate in the original equation catches this immediately: substitute x = 10.5 into −4|11 − 21| + 20 and you get −4(10) + 20 = −20, which is not zero, so 10.5 fails the original − 20 = 0 version. Only the plus-signed version accepts it.
Red-flag warnings worth checking before you commit to an answer:
Dividing by a negative coefficient without tracking the sign. Dropping the bars before isolating the absolute value alone. Reporting a single root when the isolated constant is positive. Accepting a negative right side as a valid absolute value. Rounding a fractional root to the nearest integer. Forgetting to substitute candidates back into the original equation.
One professional caveat: many online solvers accept the equation string as typed and return the empty set, while a reworded version returns 0.5 and 10.5. The solver is not wrong; the string is. Always isolate by hand first to confirm which version you are actually solving.
Practical Actions: A Fixed Five-Step Routine
Any absolute-value equation of this shape yields to the same sequence. Follow it in order and the sign errors disappear.
Step 1: move every term that does not contain the absolute value to the opposite side. For −4|11 − 2x| − 20 = 0, add 20 to get −4|11 − 2x| = 20.
Step 2: divide by the coefficient outside the bars, keeping the sign. Divide by −4 to get |11 − 2x| = −5.
Step 3: test the isolated constant. If it is negative, stop and write no real solution. If it is zero, solve the inside expression equal to zero for one root. If it is positive, continue.
Step 4: for a positive constant c, write two equations, inside = c and inside = −c. For the plus-signed variant, 11 − 2x = 5 gives x = 3, and 11 − 2x = −5 gives x = 8.
Step 5: substitute both roots into the original equation. Both must return a true statement. Then, if a greatest value is requested, report the larger root.
Threshold to remember in one line: isolated constant positive means two roots, zero means one, negative means none. That single comparison replaces most of the algebra checking.
Limits and the Role of Equation Form
The conclusion is bounded by the equation as written. Change the sign of the constant, the sign of the coefficient, or the order of terms and the solution set changes. The number −20 on the same side as the absolute-value term, combined with −4 outside the bars, forces the isolated constant negative and leaves no real solution. The same numbers rearranged with +20 on the opposite side produce two roots, whose greatest is whichever is larger.
What cannot be concluded from the stripped string alone is which form a test intended. Many calculators and answer keys publish "x = 10.5" for a problem whose displayed equation actually reads − 20 = 0, which contradicts the algebra. If your assignment asks for the greatest value among solutions, verify that the equation does in fact have solutions before computing one.
For the version with roots, the greatest value is 10.5 because it exceeds 0.5 by exactly 10, the distance spanned by the interval between them. That ten-unit separation is twice the inside constant of 5, another structural regularity: roots of |inside| = c sit 2c / (coefficient of x) apart, here 10 / 2 = 5 scaled by the sign, giving a separation of 10.
This explainer is general educational mathematics, not a personalized tutoring plan. If you are preparing for an exam or working through graded material, confirm the exact equation form with your instructor or textbook before relying on any single answer.