Inequalities practice
Solve and graph ten compound inequalities. Choose addition and subtraction, multiplication and division, or mixed operations, then connect AND intervals and OR rays with a separate graphical answer key.
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Solve each comparison, combine the conditions using AND or OR, then write the simplified solution and graph all values that satisfy it.
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First solve each inequality. AND keeps their overlap; OR includes everything in either solution. A chain such as −4 ≤ x + 2 ≤ 8 is an AND statement: subtract 2 from all parts to obtain −6 ≤ x ≤ 6.
For x < −1 or x > 3, the region between −1 and 3 satisfies neither condition. Draw an open circle at each boundary and shade outward in both directions. Arrows show that the rays continue beyond the printed window.
These ten-task worksheets use integer boundaries, positive multipliers or divisors, four bounded AND intervals and six OR pairs of rays. Each student task includes a blank number line and a symbolic response line. Separate keys provide both the simplified solution and its graph.
AND takes the intersection: a value must satisfy both conditions. OR takes the union: satisfying either condition is enough. The AND questions give bounded intervals, and the OR questions give two disjoint outward rays.
A chain such as −8 < 2x < 6 means −8 < 2x AND 2x < 6. Divide all three parts by 2 to obtain −4 < x < 3, then graph the open interval between the boundaries.
Addition/subtraction uses additive expressions. Multiplication/division uses positive multiples and fractions with positive divisors. Mixed mode includes both and combines different operations within some tasks. Each sheet has ten tasks: four AND questions and six OR questions.
Worksheets use integer boundaries and matching endpoint styles within each task. Strict comparisons use two open circles; inclusive comparisons use two filled circles.
No. The graph describes all real values in the solution set, including values between labeled ticks. The task uses the same number-line scale in its student and answer editions.
Learning objective
Combine the solution sets of two inequalities using intersection for AND and union for OR.
Worked example
Divide the first comparison by 2 to get x > −3. Subtract 4 in the second to get x < 5. Together: −3 < x < 5.
AND requires both conditions, so keep their overlap. Draw a segment from −3 to 5 with an open circle at each endpoint. For example, x = 0 satisfies both comparisons; x = 6 satisfies only the first and is excluded.
Teaching notes
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Choose addition/subtraction, multiplication/division or mixed operations. Each set has four AND tasks and six OR tasks, integer boundaries and matching endpoint inclusion within each task. Positive multipliers and divisors keep the operation choices focused; mixed tasks can combine different operations in one prompt.
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