Number line & inequalities

<p class="paragraph-header">Satisfying both conditions</p>
<p>Solve each inequality separately, then retain only the values that satisfy both. This common set is the intersection.</p>
<p>For&nbsp;\(x&gt;1\) and \(x\leq4\), the common values are \(1&lt;x\leq4\). Check each boundary against both original conditions.</p>
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<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Conditions</strong></th><th style="text-align: center; --cell-justify: center"><strong>Common solution</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(x\geq2\) and \(x\leq2\)</td><td style="text-align: center; --cell-justify: center">\(x=2\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(x\geq2\) and \(x&lt;2\)</td><td style="text-align: center; --cell-justify: center">No solution</td></tr><tr><td style="text-align: center; --cell-justify: center">\(x&gt;2\) and \(x\leq6\), with \(x\) an integer</td><td style="text-align: center; --cell-justify: center">\(x=3,4,5,6\)</td></tr></tbody></table></div></div>
<p>An endpoint is included in the common solution only if its value satisfies both conditions. A real interval may contain many values but no integers. State “no solution” when the two conditions cannot hold together.</p>

Solve each inequality separately, then retain only the values that satisfy both. This common set is the intersection.

For \(x>1\) and \(x\leq4\), the common values are \(1<x\leq4\). Check each boundary against both original conditions.

An endpoint is included in the common solution only if its value satisfies both conditions. A real interval may contain many values but no integers. State “no solution” when the two conditions cannot hold together.

Common solutions

Common solutions

Conditions	Common solution
\(x\geq2\) and \(x\leq2\)	\(x=2\)
\(x\geq2\) and \(x<2\)	No solution
\(x>2\) and \(x\leq6\), with \(x\) an integer	\(x=3,4,5,6\)