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You’re using v-bind:style wrong, use:
<p v-bind:style="{ backgroundImage: 'url(' + $IMG_PATH + '/carousel1.jpeg)' }"></p>
Example:
Vue.prototype.$IMG_PATH = 'https://hips.hearstapps.com/hmg-prod.s3.amazonaws.com/images/701'
new Vue({
el: "#app"
})
p {
height: 300px;
}
<script src="https://cdn.jsdelivr.net/npm/vue"></script>
<div id="app">
<p v-bind:style="{ backgroundImage: 'url(' + $IMG_PATH + '/beer-main-0-1496757601.jpg)' }">foobar</p>
</div>
Source:stackexchange.com