KCSE 2025 Maths P1 Q4 — Solving a Double Inequality

KCSE 2025 Form 4 Algebra

Published

The Question

“Solve the double inequality where negative one is less than or equal to the fraction (5 minus 2x) over 3, which in turn is less than 2x minus 1. Give the answer as a single combined inequality.”

1

Split the double inequality

A double inequality traps the middle expression between two others, so it is really two conditions that must hold at the same time. The reliable plan is to break it into a left inequality and a right inequality, solve each one on its own, and then combine the two answers at the end.

−1≤5−2x3<2x−1-1 \le \frac{5 - 2x}{3} < 2x - 1
−1≤5−2x3and5−2x3<2x−1-1 \le \frac{5 - 2x}{3} \quad \text{and} \quad \frac{5 - 2x}{3} < 2x - 1
2

Solve the left inequality

Clear the fraction by multiplying both sides by 3. Because 3 is positive, the direction of the inequality does not change. Then subtract 5 from both sides to isolate the x term. Dividing by negative 2 is the key move: dividing by a negative number forces you to flip the inequality sign, which turns the result into x less than or equal to 4.

−3≤5−2x-3 \le 5 - 2x
−8≤−2x-8 \le -2x
4≥x⇒x≤44 \ge x \quad \Rightarrow \quad x \le 4

Only multiplying or dividing by a negative flips the sign; adding or subtracting never does.

3

Solve the right inequality

Multiply both sides by 3 again, keeping the sign because 3 is positive, and expand the bracket on the right. Bring the x terms together by adding 2x to both sides, then add 3 to both sides to isolate the x term. Finally divide by 8, which is positive, so the sign stays as it is, giving 1 less than x.

5−2x<3(2x−1)5 - 2x < 3(2x - 1)
5−2x<6x−35 - 2x < 6x - 3
5<8x−35 < 8x - 3
8<8x⇒1<x8 < 8x \quad \Rightarrow \quad 1 < x
4

Combine the two results

The answer must satisfy both conditions together: x greater than 1 and x less than or equal to 4. On a number line, x greater than 1 is an open circle at 1 (1 is not included) while x less than or equal to 4 is a solid dot at 4 (4 is included). The solution is the overlap, everything between them, written as a single combined inequality.

1<x≤41 < x \le 4

Final Result

The combined solution is 1 < x ≤ 4, meaning x is any value greater than 1 and up to and including 4.

Why this method works

The method works because a double inequality is just shorthand for two separate statements about the same expression, both of which must be true. Solving each part independently finds the full set of x-values allowed by that part, and taking the overlap keeps only the values that obey both at once. The flip of the sign when dividing by negative 2 is essential: multiplying a true inequality by a negative reverses the order of the numbers, so the relation must reverse too, otherwise the wrong half of the number line would be selected.

✓

Test x = 2: (5 - 4)/3 = 1/3, and -1 ≤ 1/3 < 3 is true; test x = 4: (5 - 8)/3 = -1, and -1 ≤ -1 < 7 is true, so the endpoints behave as expected.