Alliance Form 3 P1 Q14 — Solving Compound Linear Inequalities
Published
The Question
“Solve the inequality where x minus 5 is less than or equal to 3x minus 8, which in turn is less than 2x minus 3, and show the solution on a number line.”
Split the compound inequality
The statement joins two inequalities into one line, so it is really two separate conditions that must both hold at the same time. Break it apart at the middle expression and solve each half on its own, then combine the results at the end.
Solve the left part
Collect the x terms on one side and the numbers on the other. Subtracting 3x from both sides leaves a negative coefficient of x, which sets up the key move in the next step.
Divide by a negative and flip the sign
To get x on its own you divide both sides by negative 2. Whenever you multiply or divide an inequality by a negative number, the direction of the sign reverses, so the less-than-or-equal becomes greater-than-or-equal.
Forgetting to flip the sign here is the most common mistake in this topic.
Solve the right part
Do the same collecting for the second inequality. This time the x term ends up positive, so no flipping is needed when you finish.
Combine and graph on a number line
Put the two results back together: x is at least 1.5 and at the same time less than 5. On the number line use a filled circle at 1.5 because x can equal it, an open circle at 5 because x cannot equal it, and shade the region between them.
Final Result
The solution is 1.5 ≤ x < 5. On the number line this is a filled (closed) circle at 1.5, an open circle at 5, and the segment between them shaded.
Why this method works
A compound inequality is a shorthand for two conditions that must be true together, so solving each half separately and then taking the overlap gives every value that satisfies both. The sign flip when dividing by a negative is not an arbitrary rule: multiplying by a negative reverses the order of numbers on the number line, so if one side was smaller before, it becomes larger after, and the inequality must be rewritten to stay true.
Test x = 3: the left gives 3-5 = -2 which is ≤ 3(3)-8 = 1, and 1 < 2(3)-3 = 3, so both parts hold — 3 lies inside 1.5 ≤ x < 5 as expected.