Inequalities

Compound Inequalities

Solve a compound inequality of the form a < bx + c ≤ d by applying the same operation to all three parts simultaneously. Because the coefficient is positive, no direction flip is needed.

compound-inequality Apply each operation to ALL THREE parts of the compound inequality at once. The two relation symbols are preserved throughout because the divisor is positive.

Example

Apply the same operation to all three parts of a compound inequality.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

−1<2x+1≤7-1 < 2x + 1 \le 7
Start with lower bound, middle expression, and upper bound.lower-1<middle2x + 1≤upper7subtractsubtract 1 from lower, middle, and upper--divide all parts by positive 2---13number lineshade betweenStart: -1 < 2x + 1 ≤ 7

Step 2 — Subtract all parts

Subtract 1 from all three parts.

−1−1<2x≤7−1-1 - \hl{1} < 2x \le 7 - \hl{1}
Subtract 1 from all three parts; both signs stay the same.lower-1<middle2x + 1≤upper7subtract-1 - 1 < 2x ≤ 7 - 1signs stay--divide all parts by positive 2---13number lineshade betweenStart: -1 < 2x + 1 ≤ 7

Step 3 — Simplify

Simplify the bounds to -2 and 6.

-2<2x≤6\hl{-2} < 2x \le \hl{6}
Simplify to -2 < 2x ≤ 6.lower-1<middle2x + 1≤upper7subtract-1 - 1 < 2x ≤ 7 - 1signs stay-2 < 2x ≤ 6divide all parts by positive 2---13number lineshade betweenStart: -1 < 2x + 1 ≤ 7

Step 4 — Divide all parts

Divide all three parts by positive 2.

−2/2<x≤6/2-2 / \hl{2} < x \le 6 / \hl{2}
Divide all three parts by positive 2.lower-1<middle2x + 1≤upper7subtract-1 - 1 < 2x ≤ 7 - 1signs stay-2 < 2x ≤ 6-2 / 2 < x ≤ 6 / 2---13number lineshade betweenStart: -1 < 2x + 1 ≤ 7

Step 5 — Solution

The solution bounds are -1 and 3.

-1<x≤3\hl{-1} < x \le \hl{3}
Solution: -1 < x ≤ 3.lower-1<middle2x + 1≤upper7subtract-1 - 1 < 2x ≤ 7 - 1signs stay-2 < 2x ≤ 6-2 / 2 < x ≤ 6 / 2-1 < x ≤ 3-13number lineshade betweenStart: -1 < 2x + 1 ≤ 7

Step 6 — Interval

Write the compound interval.

x∈(−1,3]x\in( -1 , 3 ]
Graph the interval (-1, 3].lower-1<middle2x + 1≤upper7subtract-1 - 1 < 2x ≤ 7 - 1signs stay-2 < 2x ≤ 6-2 / 2 < x ≤ 6 / 2-1 < x ≤ 3-13open at -1; closed at 3(-1, 3]Start: -1 < 2x + 1 ≤ 7